The Girth of Cayley graphs of Sylow 2-subgroups of symmetric groups $S_{2^n}$ on diagonal bases

TL;DR

This paper investigates the girth of Cayley graphs of Sylow 2-subgroups of symmetric groups on diagonal bases, providing all possible girth values, a criterion for girth 4, and bounds on non-isomorphic graph counts.

Contribution

It determines all girth values of these Cayley graphs, introduces a criterion for girth 4, and estimates the number of non-isomorphic graphs based on diagonal bases.

Findings

All possible girth values are calculated.

A criterion for girth equal to 4 is established.

A lower bound for the number of non-isomorphic Cayley graphs is proposed.

Abstract

A diagonal base of a Sylow 2-subgroup $P_n(2)$ of symmetric group $S_{2^n}$ is a minimal generating set of this subgroup consisting of elements with only one non-zero coordinate in the polynomial representation. For different diagonal bases Cayley graphs of $P_n(2)$ may have different girths (i.e. minimal lengths of cycles) and thus be non-isomorphic. In presented paper all possible values of girths of Cayley graphs of $P_n(2)$ on diagonal bases are calculated. A criterion for whenever such Cayley graph has girth equal to 4 is presented. A lower bound for the number of different non-isomorphic Cayley graphs of $P_n(2)$ on diagonal bases is proposed.