On normalized distance Laplacian eigenvalues of graphs and applications to graphs defined on groups and rings

The normalized distance Laplacian matrix of a connected graph G, denoted by D^L (G), is defined by D^L (G) = T r(G)^−1/2 D^L (G)T r(G)^−1/2, where D(G) is the distance matrix, the D^L (G) is the distance Laplacian matrix and T r(G) is the diagonal matrix of vertex transmissions of G. The set of all eigenvalues of D^L (G) including their multiplicities is the normalized distance Laplacian spectrum or D^L-spectrum of G. In this paper, we find the D^L-spectrum of the joined union of regular graphs in terms of the adjacency spectrum and the spectrum of an auxiliary matrix. As applications, we determine the D^L-spectrum of the graphs associated with algebraic structures. In particular, we find the D^L-spectrum of the power graphs of groups, the D^L-spectrum of the commuting graphs of non-abelian groups and the D^L-spectrum of the zero-divisor graphs of commutative rings. Several open problems are given for further work.