An independent set in a graph G is a set of pairwise non-adjacent vertices.Let ik(G)denote the number of independent sets of cardinality k in G.Then its generating function I(G;x)=∑^(α(G))^(k=0)ik(G)x^(k),is called ...An independent set in a graph G is a set of pairwise non-adjacent vertices.Let ik(G)denote the number of independent sets of cardinality k in G.Then its generating function I(G;x)=∑^(α(G))^(k=0)ik(G)x^(k),is called the independence polynomial of G(Gutman and Harary,1983).In this paper,we introduce a new graph operation called the cycle cover product and formulate its independence polynomial.We also give a criterion for formulating the independence polynomial of a graph.Based on the exact formulas,we prove some results on symmetry,unimodality,reality of zeros of independence polynomials of some graphs.As applications,we give some new constructions for graphs having symmetric and unimodal independence polynomials.展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.11971206,12022105)Natural Science Foundation for Distinguished Young Scholars of Jiangsu Province(Grant No.BK20200048)。
文摘An independent set in a graph G is a set of pairwise non-adjacent vertices.Let ik(G)denote the number of independent sets of cardinality k in G.Then its generating function I(G;x)=∑^(α(G))^(k=0)ik(G)x^(k),is called the independence polynomial of G(Gutman and Harary,1983).In this paper,we introduce a new graph operation called the cycle cover product and formulate its independence polynomial.We also give a criterion for formulating the independence polynomial of a graph.Based on the exact formulas,we prove some results on symmetry,unimodality,reality of zeros of independence polynomials of some graphs.As applications,we give some new constructions for graphs having symmetric and unimodal independence polynomials.
