The Main Eigenvalues of the Undirected Power Graph of a Group

Document Type : Research Paper

Authors

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran.

10.29252/asta.4.1.19

Abstract

The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $\lambda$ of $A$ is a main eigenvalue if the eigenspace $\epsilon(\lambda)$ has an eigenvector $X$ such that $X^{t}\jj\neq 0$, where $\jj$ is the all-one vector. In this paper we want to focus on the power graph of the finite cyclic group $\mathbb{Z}_{n}$ and find a condition on n where $P(\mathbb{Z}_{n})$ has exactly one main eigenvalue. Then we calculate the number of main eigenvalues of $P(\mathbb{Z}_{n})$ where $n$ has a unique prime decomposition $n = p^{r} p_2$. We also formulate a conjecture on the number of the main eigenvalues of $P(\mathbb{Z}_{n})$ for an arbitrary positive integer $n$.

Keywords


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