A family of non-Cayley cores based on vertex-transitive or strongly regular self-complementary graphs

TL;DR

This paper investigates the properties of complementary prisms of vertex-transitive or strongly regular self-complementary graphs, revealing their automorphism groups, Hamiltonian connectivity, and conditions under which they are cores, thus advancing understanding of their structural symmetries.

Contribution

It characterizes the automorphism groups, Hamiltonian connectivity, and core properties of complementary prisms based on specific classes of self-complementary graphs, providing new insights into their symmetries and structure.

Findings

Automorphism group ratios are limited to 1, 2, 4, and 12.

Complementary prisms are vertex-transitive iff the original graph is vertex-transitive and self-complementary.

Complementary prisms are cores when the original graph is strongly regular and self-complementary.

Abstract

Given a finite simple graph $\Gamma$ on $n$ vertices its complementary prism is the graph $\Gamma\bar{\Gamma}$ that is obtained from $\Gamma$ and its complement $\bar{\Gamma}$ by adding a perfect matching, where each its edge connects two copies of the same vertex in $\Gamma$ and $\bar{\Gamma}$. It generalizes the Petersen graph, which is obtained if $\Gamma$ is the pentagon. The automorphism group of $\Gamma\bar{\Gamma}$ is described for arbitrary graph $\Gamma$. In particular, it is shown that the ratio between the cardinalities of the automorphism groups of $\Gamma\bar{\Gamma}$ and $\Gamma$ can attain only values $1$, $2$, $4$, and $12$. It is shown that the Cheeger number of $\Gamma\bar{\Gamma}$ equals either 1 or $1-\frac{1}{n}$, and the two corresponding classes of graphs are fully determined. It is proved that $\Gamma\bar{\Gamma}$ is vertex-transitive if and only if $\Gamma$ is vertex-transitive and self-complementary. In this case the complementary prism is Hamiltonian-connected whenever $n>5$, and is not a Cayley graph whenever $n>1$. The main results involve endomorphisms of graph $\Gamma\bar{\Gamma}$. It is shown that $\Gamma\bar{\Gamma}$ is a core, i.e. all its endomorphisms are automorphisms, whenever $\Gamma$ is strongly regular and self-complementary. The same conclusion is obtained for many vertex-transitive self-complementary graphs. In particular, it is shown that if there exists a vertex-transitive self-complementary graph $\Gamma$ such that $\Gamma\bar{\Gamma}$ is not a core, then $\Gamma$ is neither a core nor its core is a complete graph.