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Spectrum of graphs obtained by operations. (English) Zbl 1440.05136

Summary: The distance signless Laplacian matrix of a simple connected graph \(G\) is defined as \(D^Q(G)=\text{Tr}(G)+D(G)\), where \(D(G)\) is the distance matrix of \(G\) and \(\text{Tr}(G)\) is the diagonal matrix whose main diagonal entries are the vertex transmissions in \(G\). In this paper, we first determine the distance signless Laplacian spectrum of the graphs obtained by generalization of the join and lexicographic product graph operations (namely joined union) in terms of their adjacency spectrum and the eigenvalues of an auxiliary matrix, determined by the graph \(G\). As an application, we show that new pairs of auxiliary equienergetic graphs can be constructed by joined union of regular graphs.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C12 Distance in graphs
05C76 Graph operations (line graphs, products, etc.)
05C40 Connectivity
15A18 Eigenvalues, singular values, and eigenvectors
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