Maximizing the number of independent sets in claw-free cubic graphs

Abstract

Let G be a graph. An independent set of G is a subset of vertices no two of which are connected by an edge. Denote by I(G) the set of all independent sets of G. The independence polynomial of G is I(G;λ)=∑I∈I(G)λ|I|.The cartesian product C3□K2 is commonly called the triangular-prism, it is also a generalized Petersen graph on 6 vertices and denoted by GP(3,1). In this paper, we use the occupancy method to show that for any claw-free cubic graph G,|I(G)|≤|I(GP(3,1))||V(G)|/6,furthermore, for any λ∈(0,1541],I(G;λ)≤(I(GP(3,1);λ)|V(G)|/6,with equality holding in both cases if and only if G is a disjoint union of copies of GP(3,1).