On the A_α-spectra of some join graphs

2020/08/24 by Basunia, Mainak, Mahato, Iswar, Kannan, M. Rajesh
#05C05 #05C50 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2008.10430

Abstract

Let G be a simple, connected graph and let A(G) be the adjacency matrix of G. If D(G) is the diagonal matrix of the vertex degrees of G, then for every real α∈ [0,1], the matrix A_α(G) is defined as A_α(G) = αD(G) + (1- α) A(G). The eigenvalues of the matrix A_α(G) form the A_α-spectrum of G. Let G₁ \vee G₂, G₁ \underline\vee G₂, G₁ ⟨ \textrmv ⟩ G₂ and G₁ ⟨ \textrme ⟩ G₂ denote the subdivision-vertex join, subdivision-edge join, R-vertex join and R-edge join of two graphs G₁ and G₂, respectively. In this paper, we compute the A_α-spectra of G₁ \vee G₂, G₁ \underline\vee G₂, G₁ ⟨ \textrmv ⟩ G₂ and G₁ ⟨ \textrme ⟩ G₂ for a regular graph G₁ and an arbitrary graph G₂ in terms of their A_α-eigenvalues. As an application of these results, we construct infinitely many pairs of A_α-cospectral graphs.

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