A very short proof of Sidorenko's inequality for counts of homomorphism between graphs

TL;DR

This paper presents an elementary proof of Sidorenko's inequality, demonstrating that among connected graphs with a fixed number of vertices, the star graph maximizes the number of homomorphisms into any graph.

Contribution

It offers a simple, elementary proof of a classical extremality result in graph theory, simplifying understanding of Sidorenko's inequality.

Findings

Star graphs maximize homomorphisms among connected graphs with fixed vertices.

Elementary proof simplifies previous complex demonstrations.

Reinforces the extremal property of star graphs in graph homomorphism counts.

Abstract

We provide a very elementary proof of a classical extremality result due to Sidorenko (Discrete Math. 131.1-3, 1994), which states that among all connected graphs $G$ on $k$ vertices, the $k$-vertex star maximises the number of graph homomorphisms of $G$ into any graph $H$.