Symmetric Graphs with respect to Graph Entropy

TL;DR

This paper characterizes graphs symmetric with respect to graph entropy, showing they can be uniformly covered by maximum independent sets, and proves that deciding this symmetry is co-NP-hard.

Contribution

It provides a complete characterization of symmetric graphs relative to graph entropy and establishes the computational complexity of recognizing such graphs.

Findings

A graph is symmetric with respect to graph entropy iff its vertices can be uniformly covered by maximum independent sets.

A probability distribution maximizes graph entropy iff its vertices can be uniformly covered by maximum weighted independent sets.

Deciding symmetry with respect to graph entropy is co-NP-hard.

Abstract

Let $F_G(P)$ be a functional defined on the set of all the probability distributions on the vertex set of a graph $G$. We say that $G$ is \emph{symmetric with respect to $F_G(P)$} if the uniform distribution on $V(G)$ maximizes $F_G(P)$. Using the combinatorial definition of the entropy of a graph in terms of its vertex packing polytope and the relationship between the graph entropy and fractional chromatic number, we characterize all graphs which are symmetric with respect to graph entropy. We show that a graph is symmetric with respect to graph entropy if and only if its vertex set can be uniformly covered by its maximum size independent sets. Furthermore, given any strictly positive probability distribution $P$ on the vertex set of a graph $G$, we show that $P$ is a maximizer of the entropy of graph $G$ if and only if its vertex set can be uniformly covered by its maximum weighted independent sets. We also show that the problem of deciding if a graph is symmetric with respect to graph entropy, where the weight of the vertices is given by probability distribution $P$, is co-NP-hard.