A characterization of extremal non-transmission-regular graphs by the distance (signless Laplacian) spectral radius

2024/02/01 by Jingfen Lan, Lele Liu, Lan, Jingfen +1
Computer Science · Mathematics · #05C50 #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2402.00416

openalex publication_date 2024/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a simple connected graph of order n and ∂(G) is the spectral radius of the distance matrix D(G) of G. The transmission D_i of vertex i is the i-th row sum of D(G). Denote by D_max(G) the maximum of transmissions over all vertices of G, and ∂^Q(G) is the spectral radius of the distance signless Laplacian matrix D(G)+diag(D₁,D₂,…,D_n). In this paper, we present a sharp lower bound of 2D_max(G)-∂^Q(G) among all n-vertex connected graphs, and characterize the extremal graphs. Furthermore, we give the minimum values of respective D_max(G)-∂(G) and 2D_max(G)-∂^Q(G) on trees and characterize the extremal trees.

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