The independence polynomial of a graph at -1
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2009
- DOI:
- 10.48550/arXiv.0904.4819
- arXiv:
- arXiv:0904.4819
- Bibcode:
- 2009arXiv0904.4819L
- Keywords:
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- Mathematics - Combinatorics;
- Computer Science - Discrete Mathematics;
- 05C69 (Primary);
- 05A20;
- 05C05 (Secondary)
- E-Print:
- 16 pages