The average size of independent sets of graphs

TL;DR

This paper investigates the average size of independent sets in graphs, identifying extremal cases and exploring how vertex removal affects this invariant, with specific results for trees.

Contribution

It characterizes extremal graphs for the average size of independent sets and proves existence results related to vertex removal effects.

Findings

Maximum and minimum average sizes are achieved by empty and complete graphs.

In trees, paths minimize and stars maximize the average size.

There exists a vertex whose removal does not increase the average size.

Abstract

In this paper, we study the average size of independent (vertex) sets of a graph. This invariant can be regarded as the logarithmic derivative of the independence polynomial evaluated at $1$. We are specifically concerned with extremal questions. The maximum and minimum for general graphs are attained by the empty and complete graph respectively, while for trees we prove that the path minimises the average size of independent sets and the star maximises it. While removing a vertex does not always decrease the average size of independent sets, we prove that there always exists a vertex for which this is the case.