Rajkumar, R.; Gayathri, M. Spectra of \((H_1, H_2)\)-merged subdivision graph of a graph. (English) Zbl 1427.05139 Indag. Math., New Ser. 30, No. 6, 1061-1076 (2019). Summary: In this paper, we define a ternary graph operation which generalizes the construction of subdivision graph, \(R\)-graph, central graph. Also, it generalizes the construction of overlay graph (Somodi et al., 2017), and consequently, \(Q\)-graph, total graph, and quasitotal graph. We denote this new graph by \([S(G)]_{H_2}^{H_1}\), where \(G\) is a graph and, \(H_1\) and \(H_2\) are suitable graphs corresponding to \(G\). Further, we define several new unary graph operations which becomes particular cases of this construction. We determine the adjacency and Laplacian spectra of \([S(G)]_{H_2}^{H_1}\) for some classes of graphs \(G, H_1\) and \(H_2\). From these results, we derive the \(L\)-spectrum of the graphs obtained by the unary graph operations mentioned above. As applications, these results enable us to compute the number of spanning trees and Kirchhoff index of these graphs. Cited in 8 Documents MSC: 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C76 Graph operations (line graphs, products, etc.) 05C30 Enumeration in graph theory 05C05 Trees Keywords:adjacency spectrum; Laplacian spectrum; subdivision graph; spanning trees; Kirchhoff index PDFBibTeX XMLCite \textit{R. Rajkumar} and \textit{M. Gayathri}, Indag. Math., New Ser. 30, No. 6, 1061--1076 (2019; Zbl 1427.05139) Full Text: DOI arXiv References: [1] Akbari, S.; Herman, A., Commuting decompositions of complete graphs, J. Comb. Des., 15, 2, 133-142 (2007) · Zbl 1115.05066 [2] Akbari, S.; Moazami, F.; Mohammadian, A., Commutativity of the adjacency matrices of graphs, Discrete Math., 309, 595-600 (2009) · Zbl 1194.05076 [3] Bapat, R. 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