Cayley graphs on $p$-solvable groups generated by $p$-singular elements

TL;DR

This paper investigates the spectral properties of Cayley graphs generated by p-singular elements in p-solvable groups, revealing their integrality, nullity, energy bounds, and diameter constraints.

Contribution

It applies block theory to establish integrality and nullity formulas, and provides bounds on energy and diameter for these Cayley graphs, advancing understanding of their spectral structure.

Findings

Cayley graphs are integral with nullity related to subgroup indices.

Lower bounds for graph energy are established.

Diameter of the graphs is at most the p-part of the group order.

Abstract

For a graph $\Gamma$, the multiplicity of the eigenvalue $0$, denoted by $\eta(\Gamma)$, is called the nullity of $\Gamma$. Also the energy of $\Gamma$, denoted by $\mathcal{E}(\Gamma)$, is defined as the sum of the absolute values of the eigenvalues of $\Gamma$. The index of a subgroup $H$ in a group $G$ is denoted by $[G:H]$. For a prime $p$, let $G$ be a finite $p$-solvable group whose order is divisible by $p$. Also let $\Omega_p(G)$ be the set of all $p$-singular elements of $G$. In this paper, we apply block theory of finite groups to show that the Cayley graph $\Gamma_p(G):=\mathrm{Cay}(G,\Omega_p(G))$ is an integral graph with $\eta(\Gamma_p(G))=|G|-[G:O_{p^\prime}(G)]$, where $O_{p^\prime}(G)$ is the largest normal subgroup of $G$ whose order is co-prime to $p$. We also find a lower bound for $\mathcal{E}(\Gamma_p(G))$. Finally, we prove that the diameter of $\Gamma_p(G)$ is at most $ |G|_p$.