Maximum-Size Independent Sets and Automorphism Groups of Tensor Powers of the Even Derangement Graphs

TL;DR

This paper studies the properties of tensor powers of the even derangement graph derived from the alternating group, focusing on independence, automorphisms, and other graph invariants.

Contribution

It provides a comprehensive analysis of tensor powers of the even derangement graph, including their automorphism groups and maximum independent sets.

Findings

Connectedness, diameter, and chromatic number characterized.

Maximum-size independent sets fully determined.

Automorphism groups explicitly computed.

Abstract

Let $A_n$ be the alternating group of even permutations of $X:=\{1,2,...,n\}$ and ${\mathcal E}_n$ the set of even derangements on $X.$ Denote by $A\T_n^q$ the tensor product of $q$ copies of $A\T_n,$ where the Cayley graph $A\T_n:=\T(A_n,{\mathcal E}_n)$ is called the even derangement graph. In this paper, we intensively investigate the properties of $A\T_n^q$ including connectedness, diameter, independence number, clique number, chromatic number and the maximum-size independent sets of $A\T_n^q.$ By using the result on the maximum-size independent sets $A\T_n^q$, we completely determine the full automorphism groups of $A\T_n^q.$