Hou, Yaoping; Shiu, Wai-Chee The spectrum of the edge corona of two graphs. (English) Zbl 1205.05144 Electron. J. Linear Algebra 20, 586-594 (2010). 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. Cited in 46 Documents MSC: 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C05 Trees Keywords:spectrum; adjacency matrix; Laplacian matrix; corona of graphs PDFBibTeX XMLCite \textit{Y. Hou} and \textit{W.-C. Shiu}, Electron. J. Linear Algebra 20, 586--594 (2010; Zbl 1205.05144) Full Text: DOI EuDML EMIS