[Paper Review] On some characterizations of strong power graphs of finite groups

A. K. Bhuniya, Sudip Bera|arXiv (Cornell University)|Apr 23, 2015

Graph theory and applications5 references3 citations

TL;DR

This paper characterizes the strong power graph $π_s(G)$ of finite groups, identifying when it is a line graph or Cayley graph, and computes its Laplacian spectrum and permanent. It proves that $π_s(G)$ is a Cayley graph iff $G$ is noncyclic, and is a line graph iff $G$ is cyclic of order 4, 9, or prime. Explicit formulas for the Laplacian permanent are derived for cyclic and noncyclic groups.

ABSTRACT

Let $ G $ be a finite group of order $ n$. The strong power graph $\mathcal{P}_s(G) $ of $G$ is the undirected graph whose vertices are the elements of $G$ such that two distinct vertices $a$ and $b$ are adjacent if $a^{{m}_1}$=$b^{{m}_2}$ for some positive integers ${m}_1 ,{m}_2 < n$. In this article we classify all groups $G$ for which $\mathcal{P}_s(G)$ is line graph and Caley graph. Spectrum and permanent of the Laplacian matrix of the strong power graph $\mathcal{P}_s(G)$ are found for any finite group $G$.

Motivation & Objective

• To classify finite groups $G$ for which the strong power graph $\mathcal{P}_s(G)$ is a line graph.
• To determine when $\mathcal{P}_s(G)$ is a Cayley graph, establishing a structural link to group properties.
• To compute the Laplacian spectrum of $\mathcal{P}_s(G)$ for any finite group $G$.
• To derive explicit formulas for the permanent of the Laplacian matrix of $\mathcal{P}_s(G)$, applicable to both cyclic and noncyclic groups.
• To explore spectral and graph-theoretic invariants of $\mathcal{P}_s(G)$, including algebraic connectivity and chromatic number.

Proposed method

• Uses the definition of strong power graphs: vertices are group elements, and $a \sim b$ iff $a^{m_1} = b^{m_2}$ for some $m_1, m_2 < n$.
• Applies Lemma 2.1 to characterize line graphs by excluding nine forbidden induced subgraphs.
• Analyzes the structure of $\mathcal{P}_s(\mathbb{Z}_n)$ using Euler's totient function $\phi(n)$ and non-generators to identify when the graph avoids forbidden subgraphs.
• Derives the Laplacian spectrum using spectral graph theory, leveraging the clique structure formed by non-identity elements and the identity's connections.
• Applies permanent computation techniques via generating functions and symmetric polynomial expansions, particularly for graphs with a universal vertex connected to a clique.
• Uses combinatorial identities involving binomial coefficients and signed permutations to compute the permanent of the Laplacian matrix.

Experimental results

Research questions

• RQ1 For which finite groups $G$ is the strong power graph $\mathcal{P}_s(G)$ a line graph?
• RQ2 When is $\mathcal{P}_s(G)$ isomorphic to a Cayley graph of some group?
• RQ3 What is the complete Laplacian spectrum of $\mathcal{P}_s(G)$ for any finite group $G$?
• RQ4 What is the explicit formula for the permanent of the Laplacian matrix of $\mathcal{P}_s(G)$?
• RQ5 How do spectral invariants like algebraic connectivity and chromatic number relate to the group structure?

Key findings

• The strong power graph $\mathcal{P}_s(G)$ is a line graph if and only if $G$ is cyclic of order 4, 9, or a prime.
• $\mathcal{P}_s(G)$ is a Cayley graph if and only if $G$ is noncyclic.
• For a cyclic group $G$ of order $n$, the Laplacian spectrum is fully characterized, with eigenvalues determined by the group's order and $\phi(n)$.
• The permanent of the adjacency matrix of $\mathcal{P}_s(G)$ for cyclic $G$ of order $n$ is given by a complex sum involving $\phi(n)$, $n-1$, and binomial coefficients.
• The permanent of the Laplacian matrix of $\mathcal{P}_s(G)$ for cyclic $G$ is expressed as a sum over $r$ involving $F_r(d)$, binomial coefficients, and powers of $n$ and $n-1$.
• For noncyclic $G$ of order $n$, the Laplacian matrix of $\mathcal{P}_s(G)$ is complete, and its permanent is $(-1)^n n! \left(1 - \frac{n}{1!} + \frac{n^2}{2!} - \cdots + (-1)^n \frac{n^n}{n!}\right)$.