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

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 $\mathbb{Z}_{n}.$