The independence polynomial of a graph at -1
Combinatorics
2011-01-25 v1 Discrete Mathematics
Abstract
If alpha=alpha(G) is the maximum size of an independent set and s_{k} equals the number of stable sets of cardinality k in graph G, then I(G;x)=s_{0}+s_{1}x+...+s_{alpha}x^{alpha} is the independence polynomial of G. In this paper we prove that: 1. I(T;-1) equels either -1 or 0 or 1 for every tree T; 2. I(G;-1)=0 for every connected well-covered graph G of girth > 5, non-isomorphic to C_{7} or K_{2}; 3. the absolute value of I(G;-1) is not greater than 2^nu(G), for every graph G, where nu(G) is its cyclomatic number.
Cite
@article{arxiv.0904.4819,
title = {The independence polynomial of a graph at -1},
author = {Vadim E. Levit and Eugen Mandrescu},
journal= {arXiv preprint arXiv:0904.4819},
year = {2011}
}
Comments
16 pages; 13 figures