The Cyclomatic Number of a Graph and its Independence Polynomial at −1

Abstract

If by s k is denoted the number of independent sets of cardinality k in a graph G, then $${I(G;x)=s_{0}+s_{1}x+\cdots+s_{\alpha}x^{\alpha}}$$ is the independence polynomial of G (Gutman and Harary in Utilitas Mathematica 24:97---106, 1983), where ? = ?(G) is the size of a maximum independent set. The inequality |I (G; ?1)| ≤ 2 ?(G), where ?(G) is the cyclomatic number of G, is due to (Engstrom in Eur. J. Comb. 30:429---438, 2009) and (Levit and Mandrescu in Discret. Math. 311:1204---1206, 2011). For ?(G) ≤ 1 it means that $${I(G;-1)\in\{-2,-1,0,1,2\}.}$$ In this paper we prove that if G is a unicyclic well-covered graph different from C 3, then $${I(G;-1)\in\{-1,0,1\},}$$ while if G is a connected well-covered graph of girth ? 6, non-isomorphic to C 7 or K 2 (e.g., every well-covered tree ? K 2), then I (G; ?1) = 0. Further, we demonstrate that the bounds {?2 ?(G), 2 ?(G)} are sharp for I (G; ?1), and investigate other values of I (G; ?1) belonging to the interval [?2 ?(G), 2 ?(G)].