The distance spectrum of corona and cluster of two graphs

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Abstract

Let G be a connected graph with a distance matrix D. The D-eigenvalues {μ1, μ2, . . ., . . ., μp} of G are the eigenvalues of D and form the distance spectrum or D-spectrum of G. Given two graphs G with vertex set {v1,v2,. . .. . .,vp} and H, the corona G-H is defined as the graph obtained by taking p copies of H and for each i, joining the ith vertex of G to all the vertices in the ith copy of H. Let H be a rooted graph rooted at u. Then the cluster G{H} is defined as the graph obtained by taking p copies of H and for each i, joining the ith vertex of G to the root in the ith copy of H. In this paper we describe the distance spectrum of G-H, for a connected distance regular graph G and any r-regular graph H in terms of the distance spectrum of G and adjacency spectrum of H. We also describe the distance spectrum of G{Kn}, where G is a connected distance regular graph.

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Indulal, G., & Stevanović, D. (2015). The distance spectrum of corona and cluster of two graphs. AKCE International Journal of Graphs and Combinatorics, 12(2–3), 186–192. https://doi.org/10.1016/j.akcej.2015.11.014

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