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)
We introduce a domination polynomial of a graph G. The domination polynomial of a graph G of order n 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. We obtain some properties of D(G, x) and its coefficients. Also we compute this polynomial for some… (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)
Let P(G) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent, written G∼H , if P(G) = P(H). A graph G is chromatically unique if for any graph H , G∼H implies that G is isomorphic with H. In this paper, we give the necessary and suucient conditions for a family of generalized polygon trees to be chromatically unique.