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This paper introduces two kinds of graph polynomials, clique polynomial and independent set polynomial. The paper focuses on expansions of these polynomials. Some open problems are mentioned.

Given a graph G = (V, E) and a (not necessarily proper) edge-coloring of G, we consider the complexity of finding a spanning tree of G with as many different colors as possible, and of finding one with as few different colors as possible. We show that the first problem is equivalent to finding a common independent set of maximum cardi-nality in two… (More)

A weighted graph is a graph in which each edge e is assigned a non-negative number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges. The weighted degree d w (v) of a vertex v is the sum of the weights of the edges incident with v. In this paper, we prove the following result: Suppose G is a 2-connected weighted… (More)

In this paper, some new families of integral trees with diameters 5 and 6 are constructed. All these classes are infinite. They are different from those in the existing literature. We also prove that the problem of finding integral trees of diameters 5 and 6 is equivalent to the problem of solving some Diophantine equations. The discovery of these integral… (More)

Given a graph G and a subgraph H of G, let rb(G, H) be the minimum number r for which any edge-coloring of G with r colors has a rainbow subgraph H. The number rb(G, H) is called the rainbow number of H with respect to G. Denote mK 2 a matching of size m and B n,k a k-regular bipartite graph with bipartition (X, Y) such that |X| = |Y | = n and k ≤ n. In… (More)