A note on the A_α-spectral radius of graphs

2018/05/15 by Lin, Huiqiu, Huang, Xing, Xue, Jie
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.05808

Abstract

Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For any real α∈ [0,1], Nikiforov [Merging the A- and Q-spectral theories, Appl. Anal. Discrete Math. 11 (2017) 81--107] defined the matrix A_α(G) as A_α(G)=αD(G)+(1-α)A(G). Let u and v be two vertices of a connected graph G. Suppose that u and v are connected by a path w₀(=v)w₁⋯ w_s-1w_s(=u) where d(w_i)=2 for 1≤ i≤ s-1. Let G_p,s,q(u,v) be the graph obtained by attaching the paths P_p to u and P_q to v. Let s=0,1. Nikiforov and Rojo [On the α-index of graphs with pendent paths, Linear Algebra Appl. 550 (2018) 87--104] conjectured that ρ_α(G_p,s,q(u,v))

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