Optimal Graphs for Independence and k-Independence Polynomials

Abstract

The independence polynomial I(G, x) of a finite graph G is the generating function for the sequence of the number of independent sets of each cardinality. We investigate whether, given a fixed number of vertices and edges, there exists optimally-least (optimally-greatest) graphs, that are least (respectively, greatest) for all non-negative x. Moreover, we broaden our scope to k-independence polynomials, which are generating functions for the k-clique-free subsets of vertices. For $$k \ge 3$$ , the results can be quite different from the $$k = 2$$ (i.e. independence) case.