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Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G, x) = n i=γ(G) d(G, i)x i , where d(G, i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. In this paper we study the domination polynomials of cubic graphs of order 10. As a consequence, we show that the Petersen graph is(More)
The graph consisting of s paths joining two vertices is called an s-bridge graph. In this paper, we discuss the chromaticity of some families of s-bridge graphs, especially 4-bridge graphs, and some graphs related to s-bridge graphs. 1. The chromaticity of 4-bridge graphs The graphs considered here are finite, undirected, simple and loopless. For a graph G,(More)
Let G V, E be a simple graph. A set S ⊆ V is a dominating set of G, if every vertex in V \S is adjacent to at least one vertex in S. Let P i n be the family of all dominating sets of a path P n with cardinality i, and let dP n , j |P j n |. In this paper, we construct P i n , and obtain a recursive formula for dP n , i. Using this recursive formula, we(More)