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 s_{k} is the number of independent sets of cardinality k in G, then
I(G;x)=s_{0}+s_{1}x+s_{2}x^{2}+...+s_{\alpha(G)}x^{\alpha(G)} is called the
independence polynomial of G (Gutman and Harary, 1983). If
$s_{j}=s_{\alpha-j}$, 0=< j =< alpha(G), then I(G;x) is called symmetric (or
palindromic). It is known that the graph G*2K_{1} 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*2K_{1};x)=(1+x)^{k}*I(H;x).
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