On the Maximum ABC Spectral Radius of Connected Graphs and Trees

2020/04/17 by Wenshui Lin, Yiming Zheng, Lin, Wenshui +7
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Spectral Theory (math.SP) #math.CO #math.SP

paper · pdf · doi:10.48550/arxiv.2004.08080

10 pages

arxiv created 2020/04/17 · arxiv updated 2020/04/20

Abstract

Let G=(V,E) be a connected graph, where V=\v₁, v₂, ⋯, v_n\ and m=|E|. d_i will denote the degree of vertex v_i of G, and Δ=max_{1≤ i ≤ n} d_i. The ABC matrix of G is defined as M(G)=(m_ij)_{n × n}, where m_ij=√((d_i + d_j -2)/(d_i d_j)) if v_i v_j ∈ E, and 0 otherwise. The largest eigenvalue of M(G) is called the ABC spectral radius of G, denoted by ρ_ABC(G). Recently, this graph invariant has attracted some attentions. We prove that ρ_ABC(G) ≤ √(Δ+(2m-n+1)/Δ-2). As an application, the unique tree with n ≥ 4 vertices having second largest ABC spectral radius is determined.

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