The generalized adjacency-distance matrix of connected graphs

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 combinationsS(a)(G) = a D(G) + (1 - a)A(G)where a ? [0, 1]. Let ?(S-a(G)) be the spectral radius of Sa(G). This paper presents results on S-a (G) with emphasis on ? (S-a (G)) and some results on S(G) are extended to all a in some subintervals of [0, 1]. For a ? [1/2, 1], the trees attaining the largest and the smallest ?(S-a(G)) among trees of fixed order are determined and it is proved that ?(S-a(G)) is a branching index. Moreover, for a ? (1/2, 1], the graphs that uniquely minimize ?(S-a(G)):(i) among all connected graphs of fixed order and fixed connectivity, and(ii) among all connected graphs of fixed order and fixed chromatic number are characterized.

Automated summary

This paper investigates the adjacency matrix A(G) and distance matrix D(G) defined by the connected graph G, and extends it to the adjacency-distance matrix S(G), which is further generalized as the convex linear combination S(a)(G). The focus of this study is on the spectral radius ?(S-a(G)) of S-a(G), and some results of S(G) are extended to certain subintervals of [0, 1]. For a ? [1/2, 1], the trees with the maximum and minimum ?(S-a(G)) among trees of fixed order are determined, and it is proved that ?(S-a(G)) is a branching index. Moreover, for a ? (1/2, 1], the graphs that uniquely minimize ?(S-a(G)) among all connected graphs of fixed order and fixed connectivity, as well as among all connected graphs of fixed order and fixed chromatic number, are characterized.