Homomorphisms of Strongly Regular Graphs

TL;DR

This paper proves that primitive strongly regular graphs have only automorphisms or colorings as endomorphisms, confirming a conjecture that such graphs are cores or have complete cores, using elementary linear algebra techniques.

Contribution

It establishes that homomorphisms between primitive strongly regular graphs are either isomorphisms or colorings, confirming a conjecture and providing a simple proof method.

Findings

Homomorphisms are either isomorphisms or colorings.

Primitive strongly regular graphs are cores or have complete cores.

Elementary linear algebra suffices for the proof.

Abstract

We prove that if $G$ and $H$ are primitive strongly regular graphs with the same parameters and $\varphi$ is a homomorphism from $G$ to $H$, then $\varphi$ is either an isomorphism or a coloring (homomorphism to a complete subgraph). Therefore, the only endomorphisms of a primitive strongly regular graph are automorphisms or colorings. This confirms and strengthens a conjecture of Cameron and Kazanidis that all strongly regular graphs are cores or have complete cores. The proof of the result is elementary, mainly relying on linear algebraic techniques. In the second half of the paper we discuss implications of the result and the idea underlying the proof. We also show that essentially the same proof can be used to obtain a more general statement.