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Local graph

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

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...

Vizing conjecture

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

Global parameters

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 .

Geodetic number

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 .

Vertex degree

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...

Unordered pairs representation

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].

Ulam's conjecture

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

Isomorphic graphs

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..

External path length

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 conjecture

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)

Intersection number

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

Intersection array

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 .

Strong perfect graph theorem

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).

Internal path length

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.

Integral graph

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..

Independent vertex set

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..

Edge set

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"].

Hull number

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 .

Sigma polynomial

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

Hall's condition

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 .

Shortness exponent

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

Haj&oacute;s number

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

Degree sequence

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..

Shortest path problem

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.

Graphical partition

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

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..

Graph distance matrix

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"].

Graph diameter

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"]...

Graph center

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

"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).

Connected component

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

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..

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