Extremal regular graphs: independent sets and graph homomorphisms

TL;DR

This survey reviews extremal properties of regular graphs concerning the number of independent sets and graph homomorphisms, discussing techniques, recent advances, and open problems in the field.

Contribution

It provides a comprehensive overview of methods and recent developments in understanding extremal regular graphs related to independent sets and homomorphisms.

Findings

Identification of extremal regular graphs for independent sets

Discussion of techniques for analyzing graph homomorphisms

Open problems and conjectures in extremal graph theory

Abstract

This survey concerns regular graphs that are extremal with respect to the number of independent sets, and more generally, graph homomorphisms. More precisely, in the family of of $d$-regular graphs, which graph $G$ maximizes/minimizes the quantity $i(G)^{1/v(G)}$, the number of independent sets in $G$ normalized exponentially by the size of $G$? What if $i(G)$ is replaced by some other graph parameter? We review existing techniques, highlight some exciting recent developments, and discuss open problems and conjectures for future research.