The generalized adjacency-distance matrix of connected graphs

Abstract

Let G be a connected graph with adjacency matrix A(G) and distance matrix D(G). The adjacency-distance matrix of G is defined as S(G)=D(G)+A(G). In this paper, S(G) is generalized by the convex linear combinations Sα(G)=αD(G)+(1−α)A(G)where α∈[0,1]. Let ρ(Sα(G)) be the spectral radius of Sα(G). This paper presents results on Sα(G) with emphasis on ρ(Sα(G)) and some results on S(G) are extended to all α in some subintervals of [0,1]. For α∈[1/2,1], the trees attaining the largest and the smallest ρ(Sα(G)) among trees of fixed order are determined and it is proved that ρ(Sα(G)) is a branching index. Moreover, for α∈(1/2,1], the graphs that uniquely minimize ρ(Sα(G)): among all connected graphs of fixed order and fixed connectivity, andamong all connected graphs of fixed order and fixed chromatic numberare characterized.