Sort by:

A graph is said to be locally , where is a graph (or class of graphs), when for every vertex , the graph induced on by the set of adjacent vertices of (sometimes called the ego graph in more recent literature) is isomorphic to (or to a member of) . Note that the term "neighbors" is sometimes used instead of "adjacent vertices" here (e.g., Brouwer et al. 1989), so care is needed since the definition of local graphs excludes the vertex on which a subgraph is induced, while the definitions of graph neighborhood and neighborhood graph include itself.For example, the only locally pentagonal (cycle graph ) graph is the icosahedral graph (Brouwer et al. 1989, p. 5).The following table summarizes some named graphs that have named local graphs.graphlocal graph24-cell graphcubical graphcocktail party graph 16-cell graphcocktail party graph cocktail party graph complete graph complete graph complete -partite graph complete..

LCF notation is a concise and convenient notation devised by Joshua Lederberg (winner of the 1958 Nobel Prize in Physiology and Medicine) for the representation of cubic Hamiltonian graphs (Lederberg 1965). The notation was subsequently modified by Frucht (1976) and Coxeter et al. (1981), and hence was dubbed "LCF notation" by Frucht (1976). Pegg (2003) used the notation to describe many of the cubic symmetric graphs. The notation only applies to Hamiltonian graphs, since it achieves its symmetry and conciseness by placing a Hamiltonian cycle in a circular embedding and then connecting specified pairs of nodes with edges.For example, the notation describes the cubical graph illustrated above. To see how this works, begin with the cycle graph . Beginning with a vertex , count three vertices clockwise () to and connect it to with an edge. Now advance to , count three vertices counterclockwise () to vertex , and connect and with an edge...

Let denote the domination number of a simple graph . Then Vizing (1963) conjectured thatwhere is the graph product. While the full conjecture remains open, Clark and Suen (2000) have proved the looser result

Let be a simple connected graph, and take , where is the graph diameter. Then has global parameters (respectively , ) if the number of vertices at distance (respectively, , ) from a given vertex that are adjacent to a vertex at distance from is the constant (respectively , ) depending only on (i.e., not on of ).Global parameters may be computed by the GRAPE package in GAP using the function GlobalParameters(G), which returns a list of length whose th element is the list (except that if some global parameter does not exist then is put in its place). Note that is a distance-regular graph iff this function returns no in place of a global parameter.A distance-regular graph with global parameters has intersection array .

Let denote the set of all vertices lying on an -graph geodesic in , then a set with is called a geodetic set in and is denoted .

The degree of a graph vertex of a graph is the number of graph edges which touch . The vertex degrees are illustrated above for a random graph. The vertex degree is also called the local degree or valency. The ordered list of vertex degrees in a given graph is called its degree sequence. A list of vertex degrees of a graph can be computed in the Wolfram Language using VertexDegree[g], and precomputed vertex degrees are available for particular embeddings of many named graphs via GraphData[graph, "VertexDegrees"].The minimum vertex degree in a graph is denoted , and the maximum vertex degree is denoted (Skiena 1990, p. 157).The graph vertex degree of a point in a graph, denoted , satisfieswhere is the total number of graph edges.Directed graphs have two types of degrees, knownas the indegree and the outdegree...

An unordered pair representation is a representation of an undirected graph in which edges are specified as unordered pairs of vertex indices. The unordered pairs representation of an undirected graph may be computed in the Wolfram Language using List @@@ EdgeList[g], and a graph may be constructed from an unordered pair representation using Graph[UndirectedEdge @@@ l].

Let graph have points and graph have points , where . Then if for each , the subgraphs and are isomorphic, then the graphs and are isomorphic.

Two graphs which contain the same number of graph vertices connected in the same way are said to be isomorphic. Formally, two graphs and with graph vertices are said to be isomorphic if there is a permutation of such that is in the set of graph edges iff is in the set of graph edges .Canonical labeling is a practically effective technique used for determining graph isomorphism. Several software implementations are available, including nauty (McKay), Traces (Piperno 2011; McKay and Piperno 2013), saucy, and bliss, where the latter two are aimed particularly at large sparse graphs.The equivalence or nonequivalence of two graphs can be ascertained in the Wolfram Language using the command IsomorphicGraphQ[g1, g2].Determining if two graphs are isomorphic is thought to be neither an NP-complete problem nor a P-problem, although this has not been proved (Skiena 1990, p. 181). In fact, there is a famous complexity class called graph isomorphism..

The sum over all external (square) nodes of the lengths of the paths from the root of an extended binary tree to each node. For example, in the tree above, the external path length is 25 (Knuth 1997, pp. 399-400). The internal and external path lengths are related bywhere is the number of internal nodes.

Tutte (1971/72) conjectured that there are no 3-connected nonhamiltonian bicubic graphs. However, a counterexample was found by J. D. Horton in 1976 (Gropp 1990), and several smaller counterexamples are now known.Known small counterexamples are summarized in the following table and illustrated above.namereference50Georges graphGeorges (1989), Grünbaum (2006, 2009)54Ellingham-Horton 54-graphEllingham and Horton (1983)78Ellingham-Horton 78-graphEllingham (1981, 1982)78Owens graphOwens (1983)92Horton 92-graphHorton (1982)96Horton 96-graphBondy and Murty (1976)

The intersection number of a given graph is the minimum number of elements in a set such that is an intersection graph on .

Given a distance-regular graph with integers such that for any two vertices at distance , there are exactly neighbors of and neighbors of , the sequenceis called the intersection array of .A similar type of intersection array can also be defined for a distance-transitivegraph.A distance-regular graph with global parameters has intersection array .

The theorem, originally conjectured by Berge (1960, 1961), that a graph is perfect iff neither the graph nor its graph complement contains an odd graph cycle of length at least five as an induced subgraph became known as the strong perfect graph conjecture (Golumbic 1980; Skiena 1990, p. 221). The conjecture can be stated more simply as the assertion that a graph is perfect iff it contains no odd graph hole and no odd graph antihole. The proposition can be stated even more succinctly as "a graph is perfect iff it is a Berge graph."This conjecture was proved in May 2002 following a remarkable sequence of results by Chudnovsky, Robertson, Seymour, and Thomas (Cornuéjols 2002, MacKenzie 2002).

The sum over all internal (circular) nodes of the paths from the root of an extended binary tree to each node. For example, in the tree above, the internal path length is 11 (Knuth 1997, pp. 399-400). The internal and external path lengths are related bywhere is the number of internal nodes.

An integral graph is defined as a graph whose graph spectrum consists entirely of integers. The notion was first introduced by Harary and Schwenk (1974). The numbers of simple integral graphs on , 2, ... nodes are 0, 2, 3, 6, 10, 20, 33, 71, ... (OEIS A077027), illustrated above for small .The numbers of connected simple integral graphs on , 2, ... nodes are 1, 1, 1, 2, 3, 6, 7, 22, 24, 83, ... (OEIS A064731), illustrated above for small .The following table lists common graph classes and the their members which are integral.graphintegral for of the formantiprism graph3complete graph allcycle graph 2, 3, 4, 6empty graphallprism graph3, 4, 6star graph wheel graph 4The following table lists some special named graphs that are integral and gives their spectra.graphgraph spectrum16-cell24-cellClebsch graphcubical graphcuboctahedral graphDesargues graphHall-Janko graphHoffman graphHoffman-Singleton graphLevi graphM22 graphMcLaughlin..

An independent vertex set of a graph is a subset of the vertices such that no two vertices in the subset represent an edge of . The figure above shows independent sets consisting of two subsets for a number of graphs (the wheel graph , utility graph , Petersen graph, and Frucht graph).The polynomial whose coefficients give the number of independent vertex sets of each cardinality in a graph is known as its independence polynomial.A set of vertices is an independent vertex set iff its complement forms a vertex cover (Skiena 1990, p. 218). The counts of vertex covers and independent vertex sets in a graph are therefore the same.The empty set is trivially an independent vertex setsince it contains no vertices, and therefore no edge endpoints.A maximum independent vertex set is an independent vertex set of a graph containing the largest possible number of vertices for the given graph, and the cardinality of this set is called the independence number..

The edge set of a graph is simply a set of all edges of the graph. The cardinality of the edge set for a given graph is known as the edge count of .The edge set for a particular graph embedding of a graph is given in the Wolfram Language by EdgeList[g]. The edge pairs for many named graphs can be given by the command GraphData[graph, "EdgeIndices"].

Let a set of vertices in a connected graph be called convex if for every two vertices , the vertex set of every graph geodesic lies completely in . Also define the convex hull of a graph with vertex set as the smallest convex set in containing . Then the smallest cardinality of a set whose convex hull is is called the hull number of , denoted .

Let a simple graph have vertices, chromatic polynomial , and chromatic number . Then can be written aswhere and is a falling factorial, and the polynomialis known as the -polynomial (Frucht and Giudici 1983; Li et al. 1987; Read and Wilson 1998, p. 265).-polynomials for a number of simple graphs are summarized in the following table.graph claw graph complete graph 1cubical graphcycle graph octahedral graphpath graph pentatope graph 1square graph star graph star graph tetrahedral graph 1triangle graph 1wheel graph wheel graph

Given a set , let be the set of neighbors of . Then the bipartite graph with bipartitions and has a perfect matching iff for all subsets of .

Let be the number of vertices in a graph and the length of the maximum cycle in . Then the shortness exponent of a class of graphs is defined by

The Hajós number of a graph is the maximum such that contains a subdivision of the complete graph .

Given an undirected graph, a degree sequence is a monotonic nonincreasing sequence of the vertex degrees (valencies) of its graph vertices. The number of degree sequences for a graph of a given order is closely related to graphical partitions. The sum of the elements of a degree sequence of a graph is always even due to fact that each edge connects two vertices and is thus counted twice (Skiena 1990, p. 157).The minimum vertex degree in a graph is denoted , and the maximum vertex degree is denoted (Skiena 1990, p. 157). A graph whose degree sequence contains only multiple copies of a single integer is called a regular graph. A graph corresponding to a given degree sequence can be constructed in the Wolfram Language using RandomGraph[DegreeGraphDistribution[d]].It is possible for two topologically distinct graphs to have the same degree sequence. Moreover, two distinct convex polyhedra can even have the same degree sequence for their..

The shortest path problem seeks to find the shortest path (a.k.a. graph geodesic) connecting two specific vertices of a directed or undirected graph. The length of the graph geodesic between these points is called the graph distance between and . Common algorithms for solving the shortest path problem include the Bellman-Ford algorithm and Dijkstra's algorithm.The Wolfram Language function FindShortestPath[g, u, v] can be used to find one (of possibly mutiple) shortest path between vertices and in a graph .The so-called reaching algorithm can solve the shortest path problem on an -edge graph in steps for an acyclic digraph although it allows edges to be traversed opposite their direction and given a negative length.

A partition is called graphical if there exists a graph having degree sequence . The number of graphical partitions of length is equal to the number of -node graphs that have no isolated points.The numbers of distinct graphical partitions corresponding to graphs on , 2, ... nodes are 0, 1, 2, 7, 20, 71, 240, 871, 3148, ... (OEIS A095268).A graphical partition of order is one for which the sum of degrees is . A -graphical partition only exists for even .It is possible for two topologically distinct graphs to have the same degreesequence, an example of which is illustrated above.The numbers of graphical partitions on , 4, 6, ... edges are 1, 2, 5, 9, 17, 31, 54, 90, 151, 244, ... (OEIS A000569).Erdős and Richmond (1989) showed thatand

Cospectral graphs, also called isospectral graphs, are graphs that share the same graph spectrum. The smallest pair of isospectral graphs is the graph union and star graph , illustrated above, both of which have graph spectrum (Skiena 1990, p. 85). The first example was found by Collatz and Sinogowitz (1957) (Biggs 1993, p. 12). Many examples are given in Cvetkovic et al. (1998, pp. 156-161) and Rücker et al. (2002). The smallest pair of cospectral graphs is the graph union and star graph , illustrated above, both of which have graph spectrum (Skiena 1990, p. 85).The following table summarizes some prominent named cospectral graphs.cospectral graphs126-antiprism graph, quartic vertex-transitive graph Qt1916Hoffman graph, tesseract graph16(4,4)-rook graph, Shrikhande graph2525-Paulus graphs2626-Paulus graphs28Chang graphs, 8-triangular graph70Harries graph, Harries-Wong graphDetermining..

A graphic sequence is a sequence of numbers which can be the degree sequence of some graph. A sequence can be checked to determine if it is graphic using GraphicQ[g] in the Wolfram Language package Combinatorica` .Erdős and Gallai (1960) proved that a degree sequence is graphic iff the sum of vertex degrees is even and the sequence obeys the propertyfor each integer (Skiena 1990, p. 157), and this condition also generalizes to directed graphs. Tripathi and Vijay (2003) showed that this inequality need be checked only for as many as there are distinct terms in the sequence, not for all .Havel (1955) and Hakimi (1962) proved another characterization of graphic sequences, namely that a degree sequence with and is graphical iff the sequence is graphical. In addition, Havel (1955) and Hakimi (1962) showed that if a degree sequence is graphic, then there exists a graph such that the node of highest degree is adjacent to the next highest degree..

Suppose that is a pseudograph, is the edge set of , and is the family of edge sets of graph cycles of . Then obeys the axioms for the circuits of a matroid, and hence is a matroid. Any matroid that can be obtained in this way is a graphic matroid.

Das (2018) defines the triameter of a connected graph with vertex set and vertex count at least 3 aswhere is the graph distance between vertices and .

Let denote the chromatic polynomial of a finite simple graph . Then is said to be chromatically unique if implies that and are isomorphic graphs, in other words, if is determined by its chromatic polynomial. If and are nonisomorphic but share the same chromatic polynomial, they are said to be chromatically equivalent.Cycle graphs are chromatically unique (Chao and Whitehead 1978), as are Turán graphs (Chao and Novacky 1982).Named graphs that are chromatically nonunique include the 3- and 4-barbell graph, bislit cube, bull graph, claw graph, 3-matchstick graph, Moser spindle, 2-Sierpiński sieve graph, star graphs, triakis tetrahedral graph, and 6- and 8-wheel graphs.The numbers of chromatically nonunique simple graphs on nodes for , 2, ... are 0, 0, 0, 4, 18, 115, 905, 11642, 267398, ... (OEIS A137567), while the corresponding numbers of chromatically unique graphs are 1, 2, 4, 7, 16, 41, 139, 704, 7270, ... (OEIS A137568)...

The so-called reaching algorithm can solve the shortest path problem (i.e., the problem of finding the graph geodesic between two given nodes) on an -edge graph in steps for an acyclic digraph. This algorithm allows paths such that edges traversed in the direction opposite their orientation have a negative length.No other algorithm can have better complexity because any other algorithm would have to at least examine every edge, which would itself take steps.

The thickness (or depth) (Skiena 1990, p. 251; Beineke 1997) or (Harary 1994, p. 120) of a graph is the minimum number of planar subgraphs of needed such that the graph union (Skiena 1990, p. 251).The thickness of a planar graph is therefore , and the thickness of a nonplanar graph satisfies .A lower bound for the thickness of a graph is given by(1)where is the number of edges, is the number vertices, and is the ceiling function (Skiena 1990, p. 251). The example above shows a decomposition of the complete graph into three planar subgraphs. This decomposition is minimal, so , in agreement with the bound .The thickness of a complete graph satisfies(2)except for (Vasak 1976, Alekseev and Gonchakov 1976, Beineke 1997). For , 2, ..., the thicknesses are therefore 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, ... (OEIS A124156).The thickness of a complete bipartite graph is given by(3)except possibly when and are both odd, and with there..

Two nonisomorphic graphs are said to be chromatically equivalent if they have identical chromatic polynomials. A graph that does not share a chromatic polynomial with any other nonisomorphic graph is said to be a chromatically unique graph.The chromatically equivalent simple graphs on five or fewer vertices are illustrated above.It appears to be the case that all resistance-equivalentgraphs are also chromatically equivalent.

The projective plane crossing number of a graph is the minimal number of crossings with which the graph can be drawn on the real projective plane.All graphs with crossing number 0 or 1 have projective plane crossing number 0.Richter and Siran (1996) computed the crossing number of the complete bipartite graph on an arbitrary surface. Ho (2005) showed that the projective crossing number of is given byFor , 2, ..., the first few values are therefore 0, 0, 0, 2, 4, 6, 10, 14, 18, 24, ... (OEIS A128422).

The set of graph eigenvalues of the adjacency matrix is called the spectrum of the graph. (But note that in physics, the eigenvalues of the Laplacian matrix of a graph are sometimes known as the graph's spectrum.) The spectrum of a graph with -fold degenerate eigenvalues is commonly denoted (van Dam and Haemers 2003) or (Biggs 1993, p. 8; Buekenhout and Parker 1998).The product over the elements of the spectrum of a graph is known as the characteristic polynomial of , and is given by the characteristic polynomial of the adjacency matrix of with respect to the variable .The largest absolute value of a graph's spectrum is known as its spectralradius.The spectrum of a graph may be computed in the Wolfram Language using Eigenvalues[AdjacencyMatrix[g]]. Precomputed spectra for many named graphs can be obtained using GraphData[graph, "Spectrum"].A graph whose spectrum consists entirely of integers is known as an integralgraph.The..

The characteristic polynomial is the polynomial left-hand side of the characteristicequation(1)where is a square matrix and is the identity matrix of identical dimension. Samuelson's formula allows the characteristic polynomial to be computed recursively without divisions. The characteristic polynomial of a matrix may be computed in the Wolfram Language as CharacteristicPolynomial[m, x].The characteristic polynomial of a matrix(2)can be rewritten in the particularly nice form(3)where is the matrix trace of and is its determinant.Similarly, the characteristic polynomial of a matrix is(4)where Einstein summation has been used, whichcan also be written explicitly in terms of traces as(5)In general, the characteristic polynomial has the form(6)(7)where is the matrix trace of the matrix , , and is the sum of the -rowed diagonal minors of the matrix (Jacobson 1974, p. 109).Le Verrier's algorithm for computing the characteristic..

The skewness of a graph is the minimum number of edges whose removal results in a planar graph (Harary 1994, p. 124). The skewness is sometimes denoted (Cimikowski 1992).A graph with has toroidal crossing number . (However, there exist graphs with that still have .) satisfies(1)where is the vertex count of and its edge count (Cimikowski 1992).The skewness of a disconnected graph is equalto the sum of skewnesses of its connected components.The skewness of a complete graph is given by(2)of the complete bipartite graph by(3)and of the hypercube graph by(4)(Cimikowski 1992).

A canonical labeling, also called a canonical form, of a graph is a graph which is isomorphic to and which represents the whole isomorphism class of (Piperno 2011). The complexity class of canonical labeling is not known.Efficient labeling methods yield an efficient tests for isomorphicgraphs, as provided for example by nauty, Traces, bliss, and other software implementations.

An ordered pair representation is a representation of a directed graph in which edges are specified as ordered pairs or vertex indices. The ordered pairs representation of a directed graph may be computed in the Wolfram Language using List @@@ EdgeList[g], and a graph may be constructed from an ordered pair representation using Graph[DirectedEdge @@@ l].

A bridgeless graph, also called an isthmus-free graph, is a graph that contains no graph bridges. Examples of bridgeless graphs include complete graphs on nodes, cycle graphs, the diamond graph, empty graphs, and the singleton graph.Connected bridgeless graphs are 2-edge connected and can be tested for in the Wolfram Language using KEdgeConnectedGraphQ[g, 2] or EdgeConnectivity[g] .A graph that is not bridgeless is said to be bridged.The numbers of simple bridgeless graphs on , 2, ... vertices are 1, 1, 2, 5, 16, 77, 582, 8002, ... (OEIS A263914).The numbers of simple connected bridgeless graphs on , 2, ... vertices are 1, 0, 1, 3, 11, 60, 502, 7403 ... (OEIS A007146).

A bridged graph is a graph that contains one or more graph bridges. Examples of bridged graphs include path graphs, ladder rung graphs, the bull graph, star graphs, and trees.A graph that is not bridged is said to be bridgeless. A connected bridgeless graph can be tested for in the Wolfram Language using Not[KEdgeConnectedGraphQ[g, 2]] or EdgeConnectivity[g] .The numbers of simple bridged graphs on , 2, ... vertices are 0, 1, 2, 6, 18, 79, 462, 4344, ... (OEIS A263915).The numbers of simple connected bridged graphs on , 2, ... vertices are 0, 1, 1, 3, 10, 52, 351, 3714, 63638, 1912203, ... (OEIS A052446).

The graph neighborhood of a vertex in a graph is the set of all the vertices adjacent to including itself. More generally, the th neighborhood of is the set of all vertices that lie at the distance from .The subgraph induced by the neighborhood of a graph from vertex is called the neighborhood graph.Note that while "graph neighborhood" generally includes vertices adjacent to together with the vertex itself, the term "graph neighbors" generally means vertices adjacent to excluding itself (e.g., Brouwer et al. 1989), so some care is needed when encountering these terms.

A block is a maximal biconnected subgraph of a given graph . In the illustration above, the blocks are , , and .If a graph is biconnected, then itself is called a block (Harary 1994, p. 26) or a biconnected graph (Skiena 1990, p. 175).

A shortest path between two graph vertices of a graph (Skiena 1990, p. 225). There may be more than one different shortest paths, all of the same length. Graph geodesics may be found using a breadth-first traversal (Moore 1959) or using Dijkstra's algorithm (Skiena 1990, p. 225). One (of possibly several) graph geodesics of a graph from vertex to vertex can be found in the Wolfram Language using FindShortestPath[g, u, v]. The length of the graph geodesic between these points is called the graph distance between and .The length of the maximum geodesic in a given graph is called the graph diameter, and the length of the minimum geodesic is called the graph radius.The matrix consisting of all graph distances from vertex to vertex is known as the all-pairs shortest path matrix, or more simply, the graph distance matrix...

The genus of a graph is the minimum number of handles that must be added to the plane to embed the graph without any crossings. Special cases are summarized in the following table (West 2000, p. 266).class0planar graph1toroidal graph2double-toroidal graph3pretzel graphThere are no pretzel graphs on eight or fewer vertices.Duke and Haggard (1972; Harary et al. 1973) gave a criterion for the genus of all graphs on 8 and fewer vertices. Define the double-toroidal graphs(1)(2)(3)where denotes minus the edges of . Then for any subgraph of : 1. if does not contain a Kuratowski graph (i.e., or ), 2. if contains a Kuratowski graph (i.e., is nonplanar) but does not contain any for , 3. if contains any for . The complete graph has genus(4)for , where is the ceiling function (Ringel and Youngs 1968; Harary 1994, p. 118). The values for , 2, ... are 0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 5, 6, 8, 10, ... (OEIS A000933).The complete bipartite graph has genus(5)(Ringel..

The eigenvalues of a graph are defined as the eigenvalues of its adjacency matrix. The set of eigenvalues of a graph is called a graph spectrum.The largest eigenvalue absolute value in a graph is called the spectral radius of the graph, and the second smallest eigenvalue of the Laplacian matrix of a graph is called its algebraic connectivity.

An articulation vertex of a connected graph is a vertex whose removal will disconnect the graph (Chartrand 1985). More generally, an articulation vertex is a vertex of a not-necessarily-connected graph whose removal increases the connected component count (Harary 1994, p. 26). Articulation vertices are also called cut-vertices or "cutpoints" (Harary 1994, p. 26).A graph on two or more vertices possessing no articulation vertices is called a biconnected graph. A vertex is an articulation vertex iff it appears in two biconnected components.The endpoints of a graph bridge are articulation vertices unless they both have vertex degree 1. On the other hand, it is possible for a non-bridge edge to have both endpoints be articulation vertices.The Wolfram Language function FindVertexCut[g] returns a vertex cut set of smallest size for a graph, which corresponds to an articulation vertex if the set is of size 1.The analog..

A minimum vertex cover is a vertex cover having the smallest possible number of vertices for a given graph. The size of a minimum vertex cover of a graph is known as the vertex cover number and is denoted .Every minimum vertex cover is a minimal vertex cover (i.e., a vertex cover that is not a proper subset of any other cover), but not necessarily vice versa.Finding a minimum vertex cover of a general graph is an NP-complete problem. However, for a bipartite graph, the König-Egeváry theorem allows a minimum vertex cover to be found in polynomial time.A minimum vertex cover of a graph can be computed in the Wolfram Language using FindVertexCover[g]. There is currently no Wolfram Language function to compute all minimum vertex covers.Minimum vertex covers correspond to the complements of maximumindependent vertex sets. ..

A minimum edge cover is an edge cover having the smallest possible number of edges for a given graph. The size of a minimum edge cover of a graph is known as the edge cover number of and is denoted .Every minimum edge cover is a minimal edge cover (i.e., not a proper subset of any other edge cover), but not necessarily vice versa.Only graphs with no isolated points have an edge cover (and therefore a minimum edge cover).A minimum edge cover of a graph can be computed in the Wolfram Language with FindEdgeCover[g]. There is currently no Wolfram Language function to compute all minimum edge covers of a graph.If a graph has no isolated points, thenwhere is the matching number and is the vertex count of (Gallai 1959, West 2000).

The distance between two vertices and of a finite graph is the minimum length of the paths connecting them (i.e., the length of a graph geodesic). If no such path exists (i.e., if the vertices lie in different connected components), then the distance is set equal to . In a grid graph the distance between two vertices is the sum of the "vertical" and the "horizontal" distances (right figure above).The matrix consisting of all distances from vertex to vertex is known as the all-pairs shortest path matrix, or more simply, the graph distance matrix.

The coarseness of a graph is the maximum number of edge-disjoint nonplanar subgraphs contained in a given graph . The coarseness of a planar graph is therefore .Harary (1994, pp. 121-122) gives formulas for the coarseness of and .

The circumference of a graph is the length of any longest cycle in a graph. Hamiltonian graphs on vertices therefore have circumference of .For a cyclic graph, the maximum element of the detour matrix over all adjacent vertices is one smaller than the circumference.The graph circumference of a self-complementary graph is either (i.e., the graph is Hamiltonian), , or (Furrigia 1999, p. 51).Circumferences of graphs for various classes of nonhamiltonian graphs are summarized in the table below.classcircumferenceerefbarbell graph-bishop graphbook graph 6complete bipartite graph for -cone graphgear graphgrid graph grid graph helm graph-knight graphpan graphsunlet graph web graphwheel graph -windmill graph

The matching number of graph , sometimes known as the edge independence number, is the size of a maximum independent edge set. Equivalently, it is the degree of the matching-generating polynomial(1)where is the number of -matchings of a graph . The notations , , or are sometimes also used. satisfies(2)where is the vertex count of , is the floor function. Equality occurs only for a perfect matching, and graph has a perfect matching iff(3)where is the vertex count of .The matching number of a graph is equal to the independence number of its line graph .The König-Egeváry theorem states that the matching number equals the vertex cover number (i.e., size of the smallest minimum vertex cover) are equal for a bipartite graph.If a graph has no isolated points, then(4)where is the matching number, is the size of a minimum edge cover, and is the vertex count of (West 2000).Precomputed matching numbers for many named graphs are available in the..

The bandwidth of a graph is the minimum matrix bandwidth among all possible adjacency matrices of graphs isomorphic to . Equivalently, it is the minimum graph dilation of a numbering of a graph. Bandwidth is variously denoted , , or .The bandwidth of the singleton graph is not defined, but the conventions or (Miller 1988) are sometimes adopted.The bandwidth of a disconnected graph is themaximum of the bandwidths of its connected components.The bandwidth of a connected graph satisfies the inequalities(Chinn et al. 1982), where is the vertex count of and is the graph diameter andwhere is the chromatic number.Computing the bandwidth of a graph is NP-hard.Bounds for the bandwidth of a graph have been considered by (Harper 1964), and the bandwidth of the -cube was determined by Harper (Harper 1966, Wang and Wu 2007, Harper 2010).Special cases are summarized in the following table.graphbandwidthantiprism graph4cocktail party graph complete..

Let be the number of edge covers of a graph of size . Then the edge cover polynomial is defined by(1)where is the edge count of (Akban and Oboudi 2013).Cycle graphs and complete bipartite graphs are determined by their edge cover polynomials (Akban and Oboudi 2013).The edge cover polynomial is multiplicative over graph components, so for a graph having connected components , , ..., the edge cover polynomial of itself is given by(2)The edge cover polynomial satisfies(3)where is the vertex count of a graph and is its independence polynomial (Akban and Oboudi 2013).The following table summarizes sums for the edge cover polynomials of some common classes of graphs (Akban and Oboudi 2013).graphcomplete bipartite graph complete graph cycle graph path graph The following table summarizes closed forms for the edge cover polynomials of some common classes of graphs.graphbook graph cycle graph helm graphpath graph star graph sunlet graph The following..

An automorphism of a graph is a graph isomorphism with itself, i.e., a mapping from the vertices of the given graph back to vertices of such that the resulting graph is isomorphic with . The set of automorphisms defines a permutation group known as the graph's automorphism group. For every group , there exists a graph whose automorphism group is isomorphic to (Frucht 1939; Skiena 1990, p. 185). The automorphism groups of a graph characterize its symmetries, and are therefore very useful in determining certain of its properties.The group of graph automorphisms of a graph may be computed in the Wolfram Language using GraphAutomorphismGroup[g], the elements of which may then be extracted using GroupElements. A number of software implementations exist for computing graph automorphisms, including nauty by Brendan McKay and SAUCY2, the latter of which performs several orders of magnitude faster than other implementations based on empirical..

A graph is a minor of a graph if a copy of can be obtained from via repeated edge deletion and/or edge contraction.The Kuratowski reduction theorem states that any nonplanar graph has the complete graph or the complete bipartite graph as a minor. In addition, any snark has the Petersen graph as a minor, as conjectured by Tutte (1967; West 2000, p. 304) and proved by Robertson et al. The determination of graph minors is an NP-hard problem for which no good algorithms are known, although brute-force methods such as those due to Robertson, Sanders, and Thomas exist.

An isolated point of a graph is a node of degree 0 (Hartsfield and Ringel 1990, p. 8; Harary 1994, p. 15; D'Angelo and West 2000, p. 212; West 2000, p. 22). The number of -node graphs with no isolated points are 0, 1, 2, 7, 23, 122, 888, ... (OEIS A002494), the first few of which are illustrated above. The number of graphical partitions of length is equal to the number of -node graphs that have no isolated points.Connected graphs have no isolated points.An isolated point on a curve is more commonly known as an acnode.An isolated point of a discrete set is a member of (Krantz 1999, p. 63).

Let denote the independence number of a graph . Then the Shannon capacity , sometimes also denoted , of is defined aswhere denoted the graph strong product (Shannon 1956, Alon and Lubetzky 2006). The Shannon capacity is an important information theoretical parameter because it represents the effective size of an alphabet in a communication model represented by a graph (Alon 1998). satisfiesThe Shannon capacity is in general very difficult to calculate (Brimkov et al. 2000). In fact, the Shannon capacity of the cycle graph was not determined as until 1979 (Lovász 1979), and the Shannon capacity of is perhaps one of the most notorious open problems in extremal combinatorics (Bohman 2003).Lovász (1979) showed that the Shannon capacity of the -Kneser graph is , that of a vertex-transitive self-complementary graph (which includes all Paley graphs) is , and that of the Petersen graph is 4.All graphs whose Shannon capacity is known..

Let be a simple graph with nonsingular (0,1) adjacency matrix . If all the diagonal entries of the matrix inverse are zero and all the off-diagonal entries of are nonzero, then is called a nuciferous graph (Ghorbani 2016).The path graph has adjacency matrix (and adjacency matrix inverse) given bywhich is therefore nuciferous. Initially, this was the only example known, and in fact, no others exist on 10 or fewer nodes (E. Weisstein, Mar. 18, 2016). As a result, it was conjectured by Sciriha et al. (2013) that no others exist.This conjecture was disproved by Ghorbani (2016) who found 21 Cayleygraphs examples on 24, 28, and 30 nodes.

A -matching in a graph is a set of edges, no two of which have a vertex in common (i.e., an independent edge set of size ). Let be the number of -matchings in the graph , with and the number of edges of . Then the matching polynomial is defined by(1)where vertex count of (Ivanciuc and Balaban 2000, p. 92; Levit and Mandrescu 2005) and is the matching number (which satisfies , where is the floor function).The matching polynomial is also known as the acyclic polynomial (Gutman and Trinajstić 1976, Devillers and Merino 2000), matching defect polynomial (Lovász and Plummer 1986), and reference polynomial (Aihara 1976).A more natural polynomial might be the matching-generating polynomial which directly encodes the numbers of independent edge sets of a graph and is defined by(2)but is firmly established. Fortunately, the two are related by(3)(Ellis-Monaghan and Merino 2008; typo corrected), so(4)The matching polynomial is closely..

A pair of vertices of a graph is called an -critical pair if , where denotes the graph obtained by adding the edge to and is the clique number of . The -critical pairs are never edges in . A maximal stable set of is called a forced color class of if meets every -clique of , and -critical pairs within form a connected graph.In 1993, G. Bacsó conjectured that if is a uniquely -colorable perfect graph, then has at least one forced color class. This conjecture is called the bold conjecture, and implies the strong perfect graph theorem. However, a counterexample of the conjecture was subsequently found by Sakuma (1997).

The neighborhood graph of a given graph from a vertex is the subgraph induced by the neighborhood of a graph from vertex , most commonly including itself. Such graphs are sometimes also known in more recent literature as ego graphs or ego-centered networks (Newman 2010, pp. 44-46).A graph for which the neighborhood graph at each point excluding the point itself is isomorphic to a graph is said to be a local H graph, or simply "locally ."Neighborhood graphs are implemented in the Wolfram Language as NeighborhoodGraph[g, v].

Let denote the number of nowhere-zero -flows on a connected graph with vertex count , edge count , and connected component count . This quantity is called the flow polynomial of the graph , and is given by(1)(2)where is the rank polynomial and is the Tutte polynomial (extending Biggs 1993, p. 110).The flow polynomial of a graph can be computed in the Wolfram Language using FlowPolynomial[g, u].The flow polynomial of a planar graph is related to the chromatic polynomial of its dual graph by(3)The flow polynomial of a bridged graph, and therefore also of a tree on nodes, is 0.The flow polynomials for some special classes of graphs are summarized in the table below.graphflow polynomialbook graph cycle graph ladder graph prism graph web graph0wheel graph Linear recurrences for some special classes of graphs are summarized below.graphorderrecurrenceantiprism graph4book graph 2ladder graph 1prism graph 3wheel graph 2..

The vertex count of a graph , commonly denoted or , is the number of vertices in . In other words, it is the cardinality of the vertex set.The vertex count of a graph is implemented in the Wolfram Language as VertexCount[g]. The numbers of vertices for many named graphs are given by the command GraphData[graph, "VertexCount"].

The vertex connectivity of a graph is the minimum number of nodes whose deletion disconnects it. Vertex connectivity is sometimes called "point connectivity" or simply "connectivity."A graph with is said to be connected, a graph with is said to be biconnected (Skiena 1990, p. 177), and in general, a graph with vertex connectivity is said to be -connected.Let be the edge connectivity of a graph and its minimum degree, then for any graph,(Whitney 1932, Harary 1994, p. 43).For a connected strongly regular graph or distance-regular graph with vertex degree , (A. E. Brouwer, pers. comm., Dec. 17, 2012).The vertex connectivity of a graph can be determined in the Wolfram Language using VertexConnectivity[g]. Precomputed vertex connectivities are available for many named graphs via GraphData[graph, "VertexConnecitivity"]...

The adjacency list representation of a graph consists of lists one for each vertex , , which gives the vertices to which is adjacent. The adjacency lists of a graph may be computed in the Wolfram Language using AdjacencyList[g, #]& /@ VertexList[g]and a graph may be constructed from adjacency lists using Graph[UndirectedEdge @@@ Union[ Sort /@ Flatten[ MapIndexed[{#, #2[[1]]}&, l, {2}], 1]]]

Let be the number of vertex covers of a graph of size . Then the vertex cover polynomial is defined by(1)where is the vertex count of (Dong et al. 2002).It is related to the independence polynomial by(2)(Akban and Oboudi 2013).Precomputed vertex cover polynomials for many named graphs in terms of a variable can be obtained in the Wolfram Language using GraphData[graph, "VertexCoverPolynomial"][x].The following table summarizes closed forms for the vertex cover polynomials of somecommon classes of graphs (cf. Dong et al. 2002).graphAndrásfai graph barbell graphbook graph cocktail party graphcomplete bipartite graph complete bipartite graph complete graph complete tripartite graph crown graphcycle graph empty graph gear graphhelm graphladder rung graph Möbius ladder path graph prism graphstar graph sun graphsunlet graph wheel graph Equivalent forms for the cycle graph include(3)(4)graphorderrecurrenceAndrásfai..

Let be the number of independent vertex sets of cardinality in a graph . The polynomial(1)where is the independence number, is called the independence polynomial of (Gutman and Harary 1983, Levit and Mandrescu 2005). It is also goes by several other names, including the independent set polynomial (Hoede and Li 1994) or stable set polynomial (Chudnovsky and Seymour 2004).The independence polynomial is closely related to the matching polynomial. In particular, since independent edge sets in the line graph correspond to independent vertex sets in the original graph , the matching-generating polynomial of a graph is equal to the independence polynomial of the line graph of (Levit and Mandrescu 2005):(2)The independence polynomial is also related to the clique polynomial by(3)where denotes the graph complement (Hoede and Li 1994), and to the vertex cover polynomial by(4)where is the vertex count of (Akban and Oboudi 2013).The independence..

Let be the number of dominating sets of size in a graph , then the domination polynomial of in the variable is defined aswhere is the domination number of (Kotek et al. 2012, Alikhani and Peng 2014). is multiplicative over connected components (Alikhani and Peng 2014).Precomputed dominations polynomials for many named graphs in terms of a variable and in the Wolfram Language as GraphData[graph, "DominationPolynomial"][x].The following table summarizes closed forms for the domination polynomials of some common classes of graphs (cf. Alikhani and Peng 2014).graphbarbell graphbook graph cocktail party graph complete bipartite graph complete graph empty graph helm graphsunlet graph The following table summarizes the recurrence relations for domination polynomials for some simple classes of graphs.graphorderrecurrenceantiprism graph5barbell graph3book graph 3cocktail party graph 3complete graph 2cycle graph 3empty..

A matrix for a round-robin tournament involving players competing in matches (no ties allowed) having entries(1)This scoring system differs from that used to compute a score sequence of a tournament, in which a win gives one point and a loss zero points. The matrix satisfies(2)where is the transpose of (McCarthy and Benjamin 1996).The tournament matrix for players has zero determinant iff is odd (McCarthy and Benjamin 1996). Furthermore, the dimension of the null space of an -player tournament matrix is(3)(McCarthy and Benjamin 1996).

The clique polynomial for the graph is defined as the polynomial(1)where is the clique number of , the coefficient of for is the number of cliques in a graph with vertices, and the constant term is 1 (Hoede and Li 1994, Hajiabolhassan and Mehrabadi 1998). Hajiabolhassan and Mehrabadi (1998) showed that always has a real root.The coefficient is the vertex count, is the edge count, and is the triangle count in a graph. is related to the independence polynomial by(2)where denotes the graph complement (Hoede and Li 1994).This polynomial is similar to the dependence polynomial defined as(3)(Fisher and Solow 1990), with the two being related by(4)The following table summarizes clique polynomials for some common classes of graphs.graphAndrásfai graph antiprism graphbarbell graphbook graph cocktail party graph complete bipartite graph complete graph complete tripartite graph crossed prism graphcrown graphcycle graph empty graph..

Taking a connected graph or network with a high graph diameter and adding a very small number of edges randomly, the diameter tends to drop drastically. This is known as the small world phenomenon. It is sometimes also known as "six degrees of separation" since, in the social network of the world, any person turns out to be linked to any other person by roughly six connections.Short-term memory uses small world networks between neurons to remember this sentence.In modern mathematics, the center of the network of coauthorship is considered to be P. Erdős, resulting in the so-called Erdős number. In movies, Kevin Bacon is often mentioned as the center of the movie universe, but a recent study (Reynolds) has shown Christopher Lee to be the actual center. Both actors have co-starred with Julius LeFlore, so the Lee-Bacon distance is two...

The resistance distance between vertices and of a graph is defined as the effective resistance between the two vertices (as when a battery is attached across them) when each graph edge is replaced by a unit resistor (Klein and Randić 1993, Klein 2002). This resistance distance is a metric on graphs (Klein 2002).Let be the resistance distance between vertices and in a connected graph on nodes, and define(1)where is the Laplacian matrix of and is the unit matrix. Then the resistance distance matrix is given by(2)where denotes a matrix inverse (Babić et al. 2002). This can be written explicitly as(3)Graphs that have identical resistance distance sets are known as resistance-equivalentgraphs. The smallest such pairs of graphs have nine vertices.For example, the resistance distance matrix for the tetrahedralgraph is(4)and for the cubical graph is given by(5)The resistance distances for the Platonic graphs (Klein 2002) are summarized..

Connecting the centers of touching spheres in a three-dimensional Apollonian gasket by edges given a graph known as the Apollonian network. This process is illustrated above for the case of the planar Apollonian gasket. This network turns out to have some very special properties. In addition to being either deterministic or random, they are simultaneously scale-free, display small-world effects, can be embedded in an Euclidean lattice, and show space filling as well as matching graph properties. These networks describe force chains in granular packings, fragmented porous media, hierarchical road systems, and area-covering electrical supply networks (Andrade et al. 2005). Apollonian networks share many features of neuronal systems, and have been used to study the brain (Pellegrini et al. 2007).The first few two-dimensional Apollonian networks are illustrated above. The order-twonetwork has the connectivity of the Fano plane.Apollonian..

The vertex set of a graph is simply a set of all vertices of the graph. The cardinality of the vertex set for a given graph is known as the vertex count of .The vertex set for a particular graph embedding of a graph is given in the Wolfram Language using PropertyValue[g, VertexCoordinates] or GraphEmbedding[g]. Vertex sets for many named graphs are available via GraphData[graph, "VertexCoordinates"] (for the primary embedding) and GraphData[graph, "Embeddings"] (for all available embeddings).The vertex set of an abstract simplicial complex is the union of one-point elements of (Munkres 1993, p. 15).

Let be an undirected graph, and let denote the cardinal number of the set of externally active edges of a spanning tree of , denote the cardinal number of the set of internally active edges of , and the number of spanning trees of whose internal activity is and external activity is . Then the Tutte polynomial, also known as the dichromate or Tutte-Whitney polynomial, is defined by(1)(Biggs 1993, p. 100).An equivalent definition is given by(2)where the sum is taken over all subsets of the edge set of a graph , is the number of connected components of the subgraph on vertices induced by , is the vertex count of , and is the number of connected components of .Several analogs of the Tutte polynomial have been considered for directed graphs, including the cover polynomial (Chung and Graham 1995), Gordon-Traldi polynomials (Gordon and Traldi 1993), and three-variable -polynomial (Awan and Bernardi 2016; Chow 2016). However, with the exceptions of..

The edge count of a graph , commonly denoted or and sometimes also called the edge number, is the number of edges in . In other words, it is the cardinality of the edge set.The edge count of a graph is implemented in the Wolfram Language as EdgeCount[g]. The numbers of edges for many named graphs are given by the command GraphData[graph, "EdgeCount"].

The chromatic polynomial of an undirected graph , also denoted (Biggs 1973, p. 106) and (Godsil and Royle 2001, p. 358), is a polynomial which encodes the number of distinct ways to color the vertices of (where colorings are counted as distinct even if they differ only by permutation of colors). For a graph on vertices that can be colored in ways with no colors, way with one color, ..., and ways with colors, the chromatic polynomial of is defined as the unique Lagrange interpolating polynomial of degree through the points , , ..., . Evaluating the chromatic polynomial in variables at the points , 2, ..., then recovers the numbers of 1-, 2-, ..., and -colorings. In fact, evaluating at integers still gives the numbers of -colorings.The chromatic number of a graph gives the smallest number of colors with which a graph can be colored, which is therefore the smallest positive integer such that (Skiena 1990, p. 211).For example, the cubical..

The incidence matrix of a graph gives the (0,1)-matrix which has a row for each vertex and column for each edge, and iff vertex is incident upon edge (Skiena 1990, p. 135). However, some authors define the incidence matrix to be the transpose of this, with a column for each vertex and a row for each edge. The physicist Kirchhoff (1847) was the first to define the incidence matrix.The incidence matrix of a graph (using the first definition) can be computed in the Wolfram Language using IncidenceMatrix[g]. Precomputed incidence matrices for a many named graphs are given in the Wolfram Language by GraphData[graph, "IncidenceMatrix"].The incidence matrix of a graph and adjacency matrix of its line graph are related by(1)where is the identity matrix (Skiena 1990, p. 136).For a -D polytope , the incidence matrix is defined by(2)The th row shows which s surround , and the th column shows which s bound . Incidence matrices are also..

The radius of a graph is the minimum graph eccentricity of any graph vertex in a graph. A disconnected graph therefore has infinite radius (West 2000, p. 71).Graph radius is implemented in the Wolfram Language as GraphRadius[g]. Precomputed radii for many named graphs can be obtained using GraphData[graph, "Radius"].

The periphery of a graph is the subgraph of induced by vertices that have graph eccentricities equal to the graph diameter.The periphery of a connected graph may be computed in the Wolfram Language with the command GraphPeriphery[g]. However, this function returns for disconnected graphs.

The eccentricity of a graph vertex in a connected graph is the maximum graph distance between and any other vertex of . For a disconnected graph, all vertices are defined to have infinite eccentricity (West 2000, p. 71).The maximum eccentricity is the graph diameter.The minimum graph eccentricity is called the graph radius.Eccentricities are implemented as Eccentricity[g] in the Wolfram Language package Combinatorica` . A nonstandard version of graph eccentricity for a given vertex is implemented as VertexEccentricity[g, v], which gives the eccentricity for the connected component in which is contained. Precomputed standard eccentricities (assuming infinite values for disconnected graphs) for a number of named graphs can be obtained using GraphData[graph, "Eccentricities"]...

The graph distance matrix, sometimes also called the all-pairs shortest path matrix, is the square matrix consisting of all graph distances from vertex to vertex .The mean of all distances in a (connected) graph is known as the graph's mean distance. The maximum value of all distance matrix elements is known as the graph diameter.The graph distance matrix can be computed in the Wolfram Language using the built-in function GraphDistanceMatrix[g], and precomputed distance matrices for many named graphs can be obtained using GraphData[graph, "DistanceMatrix"].

The graph diameter of a graph is the length of the "longest shortest path" (i.e., the longest graph geodesic) between any two graph vertices , where is a graph distance. In other words, a graph's diameter is the largest number of vertices which must be traversed in order to travel from one vertex to another when paths which backtrack, detour, or loop are excluded from consideration. It is therefore equal to the maximum of all values in the graph distance matrix. The above random graphs on 10 vertices have diameters 3, 4, 5, and 7, respectively.A disconnected graph has infinite diameter (West 2000, p. 71).The diameter of a graph may be computed in the Wolfram Language using GraphDiameter[g], and a fast approximation to the diameter by GraphDiameter[g, Method -> "PseudoDiameter"]. Precomputed diameters for many named graphs can be obtained using GraphData[graph, "Diameter"]...

The center of a graph is the set of vertices of graph eccentricity equal to the graph radius (i.e., the set of central points). In the above illustration, center nodes are shown in red.The center of a graph may be computed in the WolframLanguage with the command GraphCenter[g].The following table gives the number of -node simple unlabeled graphs having center nodes.OEIS, 2, ...1A0524371, 0, 1, 2, 8, 29, 180, ...2A0524380, 2, 0, 2, 4, 19, 84, ...3A0524390, 0, 3, 0, 4, 18, 119, ...4A0523400, 0, 0, 7, 0, 18, 118, ...5A0523410, 0, 0, 0, 18, 0, 129, ...60, 0, 0, 0, 0, 72, 0, ...70, 0, 0, 0, 0, 0, 414, ...

The adjacency matrix, sometimes also called the connection matrix, of a simple labeled graph is a matrix with rows and columns labeled by graph vertices, with a 1 or 0 in position according to whether and are adjacent or not. For a simple graph with no self-loops, the adjacency matrix must have 0s on the diagonal. For an undirected graph, the adjacency matrix is symmetric.The illustration above shows adjacency matrices for particular labelings of the claw graph, cycle graph , and complete graph .Since the labels of a graph may be permuted without changing the underlying graph being represented, there are in general multiple possible adjacency matrices for a given graph. In particular, the number of distinct adjacency matrices for a graph with vertex count and automorphism group order is given bywhere is the number or permutations of vertex labels. The illustration above shows the possible adjacency matrices of the cycle graph .The adjacency..

"Neighborhood" is a word with many different levels of meaning in mathematics.One of the most general concepts of a neighborhood of a point (also called an epsilon-neighborhood or infinitesimal open set) is the set of points inside an -ball with center and radius . A set containing an open neighborhood is also called a neighborhood.The graph neighborhood of a vertex in a graph is the set of all the vertices adjacent to generally including itself. More generally, the th neighborhood of is the set of all vertices that lie at the distance from . The subgraph induced by the neighborhood of a graph from vertex (again, most commonly including itself) is called the neighborhood graph (or sometimes "ego graph" in more recent literature).

A topological space decomposes into its connected components. The connectedness relation between two pairs of points satisfies transitivity, i.e., if and then . Hence, being in the same component is an equivalence relation, and the equivalence classes are the connected components.Using pathwise-connectedness, the pathwise-connected component containing is the set of all pathwise-connected to . That is, it is the set of such that there is a continuous path from to .Technically speaking, in some topological spaces, pathwise-connected is not the same as connected. A subset of is connected if there is no way to write with and disjoint open sets. Every topological space decomposes into a disjoint union where the are connected. The are called the connected components of .The connected components of a graph are the set of largest subgraphs of that are each connected. Connected components of a graph may be computed in the Wolfram Language as ConnectedComponents[g]..

Gossiping and broadcasting are two problems of information dissemination described for a group of individuals connected by a communication network. In gossiping, every person in the network knows a unique item of information and needs to communicate it to everyone else. In broadcasting, one individual has an item of information which needs to be communicated to everyone else (Hedetniemi et al. 1988).A popular formulation assumes there are people, each one of whom knows a scandal which is not known to any of the others. They communicate by telephone, and whenever two people place a call, they pass on to each other as many scandals as they know. How many calls are needed before everyone knows about all the scandals? Denoting the scandal-spreaders as , , , and , a solution for is given by , , , . The solution can then be generalized to by adding the pair to the beginning and end of the previous solution, i.e., , , , , , .Gossiping (which is also called total exchange..

Check the price

for your project

for your project

we accept

Money back

guarantee

guarantee

Price calculator

We've got the best prices, check out yourself!