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A polynomial Z_G(q,v) in two variables for abstract graphs. A graph with one graph vertex has Z=q. Adding a graph vertex not attached by any graph edges multiplies the Z by ...

The number of outward directed graph edges from a given graph vertex in a directed graph.

If a graph G has n graph vertices such that every pair of the n graph vertices which are not joined by a graph edge has a sum of valences which is >=n, then G is Hamiltonian. ...

In a network with three graph edges at each graph vertex, the number of Hamiltonian cycles through a specified graph edge is 0 or even.

A set of circuits going along the graph edges of a graph, each with an even number of graph edges, such that just one of the circuits passes through each graph vertex (Ball ...

An integral embedding of a graph, not to be confused with an integral graph, is a graph drawn such that vertices are distinct points and all graph edges have integer lengths. ...

The maximal matching-generating polynomial M_G(x) for the graph G may be defined as the polynomial M_G(x)=sum_(k=nu_L(G))^(nu(G))m_kx^k, where nu_L(G) is the lower matching ...

Grünbaum conjectured that for every m>1, n>2, there exists an m-regular, m-chromatic graph of girth at least n. This result is trivial for n=2 and m=2,3, but only a small ...

Tutte's wheel theorem states that every polyhedral graph can be derived from a wheel graph via repeated graph contraction and edge splitting. For example, the figure above ...

Isomorphic factorization colors the edges a given graph G with k colors so that the colored subgraphs are isomorphic. The graph G is then k-splittable, with k as the divisor, ...