Geodesic
Spectra of R-Vertex Join and R-Edge Join of Two Graphs
Discussiones Mathematicae. General Algebra and Applications, Tome 38 (2018) no. 1, pp. 19-31

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The R-graph R(G) of a graph G is the graph obtained from G by introducing a new vertex ue for each e ∈ E(G) and making ue adjacent to both the end vertices of e. In this paper, we determine the adjacency, Laplacian and signless Laplacian spectra of R-vertex join and R-edge join of a connected regular graph with an arbitrary regular graph in terms of their eigenvalues. Moreover, applying these results we construct some non-regular A-cospectral, L-cospectral and Q-cospectral graphs, and find the number of spanning trees.
Keywords: spectrum, cospectral graphs, R-vertex join, R-edge join 
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Das, Arpita; Panigrahi, Pratima. Spectra of R-Vertex Join and R-Edge Join of Two Graphs. Discussiones Mathematicae. General Algebra and Applications, Tome 38 (2018) no. 1, pp. 19-31. http://geodesic.mathdoc.fr/item/DMGAA_2018_38_1_a1/