IRRELEVANT EDGES FOR SOME GRAPH POLYNOMIALS

Graph polynomials are a well-developed area useful for analyzing properties of graphs. Let f be a complex-valued function defined on the set of graphs G such that G 1 ∼ G 2 implies f(G 1 ) = f(G 2 ). Let G be a graph on n vertices and S(G) be the set of all subgraphs of G. Define S k (G) = { H : H ∈ S(G) and | V (H) | = k } , p(G, k) = ∑ H ∈ S k (G) f(H). Then, the general graph polynomial of G is defined as P(G, x) = ∑ n k=0 p(G, k)x k . An irrelevant edge of P(G, x) is an edge of G, such that P(G, x) = P(G \ e, x). In this paper we consider some graph polynomials and study their irrelevant edges.