On Symmetry of Independence Polynomials
Abstract
An independent set in a graph is a set of pairwise non-adjacent vertices, and alpha(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while mu(G) is the cardinality of a maximum matching. If sk is the number of independent sets of cardinality k in G, then I(G;x)=s0+s1x+s2x2+...+sα(G)xα(G) is called the independence polynomial of G (Gutman and Harary, 1983). If sj=sα-j, 0=< j =< alpha(G), then I(G;x) is called symmetric (or palindromic). It is known that the graph G*2K1 obtained by joining each vertex of G to two new vertices, has a symmetric independence polynomial (Stevanovic, 1998). In this paper we show that for every graph G and for each non-negative integer k =< mu(G), one can build a graph H, such that: G is a subgraph of H, I(H;x) is symmetric, and I(G*2K1;x)=(1+x)k*I(H;x).