Abstract
For a simple connected graph G of order n, the normalized Laplacian is a square matrix of order n, defined as [Formula: see text], where [Formula: see text] is the diagonal matrix whose i-th diagonal entry is [Formula: see text]. In this paper, 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 [Formula: see text], the power graph [Formula: see text] of a group [Formula: see text] 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 the other. As a consequence of the joined union of graphs, we investigate the normalized Laplacian eigenvalues of the power graphs of the finite cyclic group [Formula: see text].
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