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

 Rather, Bilal A., Ganie, Hilal A. and  Aouchiche, Mustapha 

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The normalized distance Laplacian matrix of a connected graph G, denoted by D^{\mathcal{L}}(G), is defined by D^{\mathcal{L}}(G)=Tr(G)^{-1/2}D^L(G)Tr(G)^{-1/2}, where D(G) is the distance matrix, the D^{L}(G) is the distance Laplacian matrix and Tr(G) is the diagonal matrix of vertex transmissions of G. The set of all eigenvalues of D^{\mathcal{L}}(G) including their multiplicities is the normalized distance Laplacian spectrum or D^{\mathcal{L}}-spectrum of G. In this paper, we find the D^{\mathcal{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^{\mathcal{L}}-spectrum of the graphs associated with algebraic structures. In particular, we find the D^{\mathcal{L}}-spectrum of the power graphs of groups, the D^{\mathcal{L}}-spectrum of the commuting graphs of non-abelian groups and the D^{\mathcal{L}}-spectrum of the zero-divisor graphs of commutative rings. Several open problems are given for further work.


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 Ganie, Hilal A.,  Rather, Bilal A., Aouchiche, Mustapha