Various spectra and energies of subgroup generating bipartite graph

TL;DR

This paper investigates the spectral properties and energies of subgroup generating bipartite graphs for specific groups, providing new insights into their energetic classifications and confirming the E-LE conjecture for these cases.

Contribution

It computes spectra and energies of subgroup generating bipartite graphs for dihedral and dicyclic groups, and verifies the E-LE conjecture for these groups.

Findings

Determined spectra and energies for the graphs of specified groups.

Classified the graphs as hypoenergetic, hyperenergetic, and other energy types.

Confirmed the E-LE conjecture for these groups.

Abstract

Let $L(G)$ be the set of all subgroups of a group $G$. The subgroup generating bipartite graph $\mathcal{B}(G)$ defined on $G$ is a bipartite graph whose vertex set is partitioned into two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in L(G)$ are adjacent if $H$ is generated by $a$ and $b$. In this paper, we compute various spectra and energies of $\mathcal{B}(G)$ and determine whether $\mathcal{B}(G)$ is hypoenergetic, hyperenergetic, CN-hyperenergetic, L-hyperenergetic or Q-hyperenergetic if $G$ is a dihedral group of order $2p$ and $2p^2$ and dicyclic group of order $4p$ and $4p^2$, where $p$ is any prime. We also show that $\mathcal{B}(G)$ satisfies E-LE conjecture for these groups.