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Homomorphisms of strongly regular graphs. (English) Zbl 1417.05252
Summary: 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). Moreover, any such coloring is optimal for $$G$$ and its image is a maximum clique of $$H$$. Therefore, the only endomorphisms of a primitive strongly regular graph are automorphisms or colorings. This confirms and strengthens a conjecture of P. J. Cameron and P. A. Kazanidis [J. Aust. Math. Soc. 85, No. 2, 145–154 (2008; Zbl 1167.05032)] 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 the idea underlying the proof, namely that it can be seen as a straightforward application of complementary slackness to a dual pair of semidefinite programs that define the Lovász theta function. We also consider implications of the result and show that essentially the same proof can be used to obtain a more general statement. We believe that one of the main contributions of the work is the novel proof technique, which is the first able to make use of the combinatorial regularity of a graph in order to obtain results about its endomorphisms/homomorphisms. Thus we expect this approach to have further applicability to the study of homomorphisms of highly regular graphs.
MSC:
 05E30 Association schemes, strongly regular graphs 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) 90C22 Semidefinite programming 05C15 Coloring of graphs and hypergraphs
Software:
Digraphs; SageMath
Full Text:
References:
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