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A k-matching in a graph G is a set of k edges, no two of which have a vertex in common (i.e., an independent edge set of size k). Let Phi_k be the number of k-matchings of ...

Ore (1962) noted that not only does a tree possesses a unique shortest path between any two vertices, but that there also exist also other connected graphs having the same ...

Vizing's theorem states that a graph can be edge-colored in either Delta or Delta+1 colors, where Delta is the maximum vertex degree of the graph. This partitions graphs into ...

The (26,8)-Paulus graph having the largest possible graph automorphism group order of all 26-node Paulus graphs (namely 120) is sometimes known as the ...

An induced subgraph is a subgraph obtained from an original graph by removing a subset of vertices and/or edges together with any edges whose endpoints are both in this ...

A maximal clique is a clique that cannot be extended by including one more adjacent vertex, meaning it is not a subset of a larger clique. A maximum clique (i.e., clique of ...

The cubic Archimedean graph on 60 nodes and 90 edges that is the skeleton of the truncated dodecahedron. It is implemented in the Wolfram Language as ...

A Hamiltonian path, also called a Hamilton path, is a graph path between two vertices of a graph that visits each vertex exactly once. If a Hamiltonian path exists whose ...

The clique polynomial C_G(x) for the graph G is defined as the polynomial C_G(x)=1+sum_(k=1)^(omega(G))c_kx^k, (1) where omega(G) is the clique number of G, the coefficient ...

Let P(G) denote the chromatic polynomial of a finite simple graph G. Then G is said to be chromatically unique if P(G)=P(H) implies that G and H are isomorphic graphs, in ...