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The spectrum of the edge corona of two graphs. (English) Zbl 1205.05144

Summary: Given two graphs \(G_1\), with vertices \(1,2,\dots,n\) and edges \(e_1,e_2,\dots,e_m\), and \(G_2\), the edge corona \(G_1 \lozenge G_2\) of \(G_1\) and \(G_2\) is defined as the graph obtained by taking \(m\) copies of \(G_2\) and for each edge \(e_k = ij\) of \(G\), joining edges between the two end-vertices \(i,j\) of \(e_k\) and each vertex of the \(k\)-copy of \(G_2\) . In this paper, the adjacency spectrum and Laplacian spectrum of \(G_1 \lozenge G_2\) are given in terms of the spectrum and Laplacian spectrum of \(G_1\) and \(G_2\), respectively. As an application of these results, the number of spanning trees of the edge corona is also considered.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C05 Trees
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