Optimal Graphs for Independence and $k$-Independence Polynomials

TL;DR

This paper explores the existence of optimal graphs that maximize or minimize independence and $k$-independence polynomials for given vertices and edges, revealing different behaviors for various $k$ values.

Contribution

It introduces the concept of optimal graphs for independence and $k$-independence polynomials and analyzes their existence and properties across different $k$ values.

Findings

Existence of optimal graphs varies with $k$.

Different behaviors observed for $k eq 2$.

Results highlight complexities in polynomial extremal problems.

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 $k \geq 3$, the results can be quite different from the $k = 2$ (i.e. independence) case.