On normalized Laplacian eigenvalues of power graphs associated to finite cyclic groups

TL;DR

This paper derives the normalized Laplacian eigenvalues of power graphs of finite cyclic groups, linking spectral properties to group structure and graph operations.

Contribution

It provides explicit formulas for normalized Laplacian eigenvalues of power graphs of cyclic groups using graph joins and quotient matrices.

Findings

Eigenvalues expressed in terms of adjacency eigenvalues

Results for power graphs of cyclic groups

Connections between group structure and spectral properties

Abstract

For a simple connected graph $ G $ of order $ n $, the normalized Laplacian is a square matrix of order $ n $, defined as $\mathcal{L}(G)= D(G)^{-\frac{1}{2}}L(G)D(G)^{-\frac{1}{2}}$, where $ D(G)^{-\frac{1}{2}} $ is the diagonal matrix whose $ i$-th diagonal entry is $ \frac{1}{\sqrt{d_{i}}} $. In this article, we find the normalized Laplacian eigenvalues of the joined union of regular graphs in terms of the adjacency eigenvalues and the eigenvalues of quotient matrix associated with graph $ G $. For a finite group $\mathcal{G}$, the power graph $\mathcal{P}(\mathcal{G})$ of a group $ \mathcal{G} $ is defined as the simple graph in which two distinct vertices are joined by an edge if and only if one is the power of other. As a consequence of the joined union of graphs, we investigate the normalized Laplacian eigenvalues of power graphs of finite cyclic group $ \mathbb{Z}_{n}. $