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The graph categorical product, also called the tensor product, is the graph product denoted and defined by the adjacency relations ( and ).The graph categorical product is known as the bipartite double graph of .

"The" Y-graph is another term used to refer to a clawgraph.The term "Y-graph" is also used to refer to a graph expansion with the Y graph as its base (e.g., Horton and Bouwer 1991). There are exactly four graph expansions with Y-graph base that are symmetric (Biggs 1993, p. 147).graphexpansion 28Coxeter graph(7; 1, 2, 4)56cubic symmetric graph (14; 1, 3, 5)112cubic symmetric graph (28; 1, 3, 9)224cubic symmetric graph (56; 1, 9, 25)

In general, a graph product of two graphs and is a new graph whose vertex set is and where, for any two vertices and in the product, the adjacency of those two vertices is determined entirely by the adjacency (or equality, or non-adjacency) of and , and that of and . There are cases to be decided (three possibilities for each, with the case where both are equal eliminated) and thus there are different types of graph products that can be defined.The most commonly used graph products, given by conditions sufficient and necessary for adjacency, are summarized in the following table (Hartnell and Rall 1998). Note that the terminology is not quite standardized, so these products may actually be referred to by different names by different sources (for example, the graph lexicographic product is also known as the graph composition; Harary 1994, p. 21). Many other graph products can be found in Jensen and Toft (1994).Graph products can be computed using..

The Cartesian graph product , sometimes simply called "the" graph product (Beineke and Wilson 2004, p. 104) and sometimes denoted (e.g., Salazar and Ugalde 2004) of graphs and with disjoint point sets and and edge sets and is the graph with point set and adjacent with whenever or (Harary 1994, p. 22).Graph Cartesian products can be computed using the undocumented Wolfram Language function GraphComputation`GraphProduct[G1, G2, "Cartesian"].The following table gives examples of some graph Cartesian products. Here, denotes a cycle graph, a complete graph, a path graph, and a star graph.productresultgrid graph ladder graph grid graph prism graph stacked prism graph torus grid graph with circulant graph book graphstacked book graphcrown graph rook graph rook complement graphhypercube graph ..

The contraction of a pair of vertices and of a graph produces a graph in which the two nodes and are replaced with a single node such that is adjacent to the union of the nodes to which and were originally adjacent. In vertex contraction, it doesn't matter if and are connected by an edge; if they are, the edge is simply removed upon contraction (Pemmaraju and Skiena 2003, p. 231). Note that Skiena (1990, p. 91) is ambiguous about the distinction between vertex contraction and edge contraction, and confusingly refers to vertex contraction on vertices and as "contracting an edge ."The figure above shows a random graph contracted on vertices and . Vertex contraction can be implemented using Contract[g, v1, v2] in the Wolfram Language package Combinatorica` .

The th power of a graph is a graph with the same set of vertices as and an edge between two vertices iff there is a path of length at most between them (Skiena 1990, p. 229). Since a path of length two between vertices and exists for every vertex such that and are edges in , the square of the adjacency matrix of counts the number of such paths. Similarly, the th element of the th power of the adjacency matrix of gives the number of paths of length between vertices and . Graph powers are implemented in the Wolfram Language as GraphPower[g, k].The graph th power is then defined as the graph whose adjacency matrix given by the sum of the first powers of the adjacency matrix,which counts all paths of length up to (Skiena 1990, p. 230).Raising any graph to the power of its graph diameter gives a complete graph. The square of any biconnected graph is Hamiltonian (Fleischner 1974, Skiena 1990, p. 231). Mukhopadhyay (1967) has considered "square..

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 topological sort is a permutation of the vertices of a graph such that an edge implies that appears before in (Skiena 1990, p. 208). Only acyclic digraphs can be topologically sorted. The topological sort of a graph can be computed using TopologicalSort[g] in the Wolfram Language package Combinatorica` .

Given a planar graph , its geometric dual is constructed by placing a vertex in each region of (including the exterior region) and, if two regions have an edge in common, joining the corresponding vertices by an edge crossing only . The result is always a planar pseudograph. However, an abstract graph with more than one embedding on the sphere can give rise to more than one dual.Whitney showed that the geometric dual graph and combinatorial dual graph are equivalent (Harary 1994, p. 115), and so may simply be called "the" dual graph.

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.

The graph product denoted and defined by the adjacency relations () or ( and ). The graph lexicographic product is also known as the graph composition (Harary 1994, p. 21).The "double graph" of a given graph is the graph lexicographic product .

Let be a (not necessarily simple) undirected edge-weighted graph with nonnegative weights. A cut of is any nontrivial subset of , and the weight of the cut is the sum of weights of edges crossing the cut. A mincut is then defined as a cut of of minimum weight. The problem is polynomial time solvable as a series of network flow problems or using the algorithm of Stoer and Wagner (1994).

The join of graphs and with disjoint point sets and and edge sets and is the graph union together with all the edges joining and (Harary 1994, p. 21). Graph joins can be computed using GraphJoin[G1, G2] in the Wolfram Language package Combinatorica` .A complete -partite graph is the graph join of empty graphs on , , ... nodes. A wheel graph is the join of a cycle graph and the singleton graph. Finally, a star graph is the join of an empty graph and the singleton graph (Skiena 1990, p. 132).The following table gives examples of some graph joins. Here denotes an empty graph (i.e., the graph complement of the complete graph ), a cycle graph, and the singleton graph.productresultcomplete -partite graph wheel graph star graph cone graph fan graph

Let be a (not necessarily simple) undirected edge-weighted graph with nonnegative weights. A cut of is any nontrivial subset of , and the weight of the cut is the sum of weights of edges crossing the cut. A maxcut is then defined as a cut of of maximum weight. Determining the maxcut of a graph is an NP-hard problem.

In a graph , contraction of an edge with endpoints is the replacement of and with a single vertex such that edges incident to the new vertex are the edges other than that were incident with or . The resulting graph has one less edge than .Graph minors are defined in terms of edge contractions.

Let be the vertex set of a simple graph and its edge set. Then a graph isomorphism from a simple graph to a simple graph is a bijection such that iff (West 2000, p. 7).If there is a graph isomorphism for to , then is said to be isomorphic to , written .There exists no known P algorithm for graph isomorphism testing, although the problem has also not been shown to be NP-complete. As a result, the special complexity class graph isomorphism complete is sometimes used to refer to the problem of graph isomorphism testing.

The set of all edge automorphisms of , denoted . Let be the line graph of a graph . Then the edge automorphism group is isomorphic to ,(Holton and Sheehan 1993, p. 26).

Let be a set and a nonempty family of distinct nonempty subsets of whose union is . The intersection graph of is denoted and defined by , with and adjacent whenever and . Then a graph is an intersection graph on if there exists a family of subsets for which and are isomorphic graphs (Harary 1994, p. 19). Graph intersections can be computed in the Wolfram Language using GraphIntersection[g, h].

An edge automorphism of a graph is a permutation of the edges of that sends edges with common endpoint into edges with a common endpoint. The set of all edge automorphisms of from a group called the edge automorphism group of , denoted .

"The" graph is the path graph on two vertices: .An -graph for and is a generalization of a generalized Petersen graph and has vertex setand edge setwhere the subscripts are read modulo (Bouwer et al. 1988, Žitnik et al. ). Such graphs can be constructed by graph expansion on .If the restriction is relaxed to allow and to equal , gives the ladder rung graph and gives the sunlet graph .Two -graphs and are isomorphic iff there exists an integer relatively prime to such that either or (Boben et al. 2005, Horvat et al. 2012, Žitnik 2012).The graph is connected iff . If , then the graph consists of copies of (Žitnik et al. 2012).The -graph corresponds to copies of the graph The following table summarizes special named -graphs and classes of named -graphs.graphcubical graph Petersen graph Dürer graphMöbius-Kantor graphdodecahedral graphDesargues graphNauru graphprism graph generalized Petersen graph All -graphs..

Given any tree having vertices of vertex degrees of 1 and 3 only, form an -expansion by taking disjoint copies of and joining corresponding leaves by an -cycle where, however, the th leaf on the th copy need not be connected to the th leaf on the st copy, but will in general be connected to the th copy. The set of values are known as the steps.The resulting graphs are always cubic, and there exist exactly 13 graph expansions that are symmetric as well, as summarized in the following table (Biggs 1993, p. 147). E. Gerbracht (pers. comm., Jan. 29, 2010) has shown that all the graphs in this table are unit-distance.graphFostergeneralized Petersen graphbase graphexpansion 8cubical graph I graph(4; 1, 1)10Petersen graphI graph(5; 1, 2)16Möbius-Kantor graphI graph(8; 1, 3)20dodecahedral graphI graph(10; 1, 2)20Desargues graphI graph(10; 1, 3)24Nauru graphI graph(12; 1, 5)28Coxeter graphY graph(7; 1, 2, 4)48cubic symmetric..

Given a planar graph , a geometric dual graph and combinatorial dual graph can be defined. Whitney showed that these are equivalent (Harary 1994), so that one may speak of "the" dual graph . The illustration above shows the process of constructing a geometric dual graph.Polyhedral graphs have unique dual graphs. While some nonpolyhedral planar graphs also have a unique dual, a general planar graph has multiple dual graphs depending on the choice of planar drawing. A planar graph has a unique embedding, and consequently a unique dual, iff it is a subdivision of a polyhedral graph. The complete bipartite graph is an example of a planar nonpolyhedral graph that is not 3-vertex-connected but whose embeddings are all isomorphic, meaning its dual graphs are also isomorphic.The dual graph of a polyhedral graph has graph vertices each of which corresponds to a face of and each of whose faces corresponds to a graph vertex of . Two nodes in are connected..

"The" H graph is the tree on 6 vertices illustrated above. It is implemented in the Wolfram Language as GraphData["HGraph"].The term "H-graph" is also used to refer to a graph expansion with the 6-vertex H graph as its base (e.g., Horton and Bouwer 1991). There are exactly two graph expansions with H-graph base that are symmetric (Biggs 1993, p. 147).graphexpansion 102Biggs-Smith graph (17; 3, 5, 6, 7)204cubic symmetric graph (34; 3, 5, 7, 11)

Let a tree be a subgraph of a cubic graph . The graph excision is the graph resulting from removing the tree, then merging the edges. For example, if in the Levi graph (left figure) the tree formed by the 6 interior points (middle figure) is excised, the McGee graph (right figure) results. Similarly, excising the Heawood graph gives the Petersen graph, and excising the generalized hexagon (i.e., the unique 12-cage graph) gives the Balaban 11-cage (Biggs 1998).The reverse of excision is insertion. Both operations are used in the analysis ofcages.The following table gives some cubic symmetricgraphs with named edge-excised graphs, illustrated above.graphedge-excised graphutility graphtetrahedral graphcubical graph3-prism graphPetersen graph4-Möbius ladderHeawood graph12-cubic graph 84Möbius-Kantor graph14-cubic graph 503dodecahedral graphcubic polyhedral graph Cp34..

The double graph of a given graph is constructed by making two copies of (including the initial edge set of each) and adding edges and for every edge of .Note that the double graph differs from the bipartite double graph in that the initial edge set is retained in the double graph, while it is discarded in the bipartite double graph.

The union of graphs and with disjoint point sets and and edge sets and is the graph with and (Harary 1994, p. 21; Gross and Yellen 2006, p. 85). This operation is sometimes also known explicitly as the graph disjoint union.Graph unions can be computed in the Wolfram Language using GraphDisjointUnion[g1, g2, ...]. (Note that the Wolfram Language function GraphUnion[g1, g2] performs a different operation from the usual graph union.)The graph union of copies of a graph is commonly denoted (Harary 1990, p. 21).

The graph difference of graphs and is the graph with adjacency matrix given by the difference of adjacency matrices of and . A graph difference is defined when the orders of and are the same, and can be computed in the Wolfram Language using GraphDifference[g, h].

The graph sum of graphs and is the graph with adjacency matrix given by the sum of adjacency matrices of and . A graph sum is defined when the orders of and are the same, and can be computed using GraphSum[g, h] in the Wolfram Language package Combinatorica` .

Let be the cycle rank of a graph , be the cocycle rank, and the relative complement of a subgraph of be defined as that subgraph obtained by deleting the lines of . Then a graph is a combinatorial dual of if there is a one-to-one correspondence between their sets of lines such that for any choice and of corresponding subsets of lines,where is the subgraph of with the line set .Whitney showed that the geometric dual graph and combinatorial dual graph are equivalent (Harary 1994, p. 115), and so may simply be called "the" dual graph. Also, a graph is planar iff it has a combinatorial dual (Harary 1994, p. 115).

The graph strong product is a graph product variously denoted , (Alon, and Lubetzky 2006), or (Beineke and Wilson 2004, p. 104) defined by the adjacency relations ( and ) or ( and ) or ( and ).In other words, the graph strong product of two graphs and has vertex set and two distinct vertices and are connected iff they are adjacent or equal in each coordinate, i.e., for , either or , where is the edge set of .

The composition of graphs and with disjoint point sets and and edge sets and is the graph with point vertex and adjacent with whenever or (Harary 1994, p. 22). It is also called the graph lexicographic product.

The bipartite double graph of a given graph , perhaps better called the Kronecker cover, is constructed by making two copies of the vertex set of (omitting the initial edge set entirely) and constructing edges and for every edge of . The bipartite double graph is equivalent to the graph categorical product .Note that the bipartite double differs from the plain double graph in that the initial edge set is discarded in the bipartite double graph, while it is retained in the double graph.The following table summarizes bipartite double graphs for some named graphs and classes of graphs.bipartite double of 16-cell graphHaar graph 4-antiprism graphquartic vertex-transitive graph Qt485-antiprism graphHaar graph Biggs-Smith graphcubic symmetric graph Clebsch graphhypercube graph complete graph crown graph Coxeter graphcubic symmetric graph cubic symmetric graph cubic symmetric graph cubic symmetric graph cubic symmetric graph cubic symmetric..

An orientation of an undirected graph is an assignment of exactly one direction to each of the edges of . Only connected, bridgeless graphs can have a strong orientation (Robbins 1939; Skiena 1990, p. 174). An oriented complete graph is called a tournament.

A line graph (also called an adjoint, conjugate, covering, derivative, derived, edge, edge-to-vertex dual, interchange, representative, or -obrazom graph) of a simple graph is obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge iff the corresponding edges of have a vertex in common (Gross and Yellen 2006, p. 20).The line graph of a directed graph is the directed graph whose vertex set corresponds to the arc set of and having an arc directed from an edge to an edge if in , the head of meets the tail of (Gross and Yellen 2006, p. 265).Line graphs are implemented in the Wolfram Language as LineGraph[g]. Precomputed line graph identifications of many named graphs can be obtained in the Wolfram Language using GraphData[graph, "LineGraphName"].The numbers of simple line graphs on , 2, ... vertices are 1, 2, 4, 10, 24, 63, 166, 471, 1408, ... (OEIS A132220), and the numbers of connected..

The complement of a graph , sometimes called the edge-complement (Gross and Yellen 2006, p. 86), is the graph , sometimes denoted or (e.g., Clark and Entringer 1983), with the same vertex set but whose edge set consists of the edges not present in (i.e., the complement of the edge set of with respect to all possible edges on the vertex set of ). The graph sum on a -node graph is therefore the complete graph , as illustrated above.A graph complement can be computed in the WolframLanguage by the command GraphComplement[g].

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