On the Spectral Radius of the Maximum Degree Matrix of Graphs

Let G be a graph with n vertices, and let d G ( u ) denote the degree of vertex u in G . The maximum degree matrix M G of G is the square matrix of order n whose ( u , v ) -entry is equal to max d G ( u ) , d G ( v ) if vertices u and v are adjacent in G , and zero otherwise. Let B p , q , r be the graph obtained from the complete graph K p by removing an edge u v , and identifying vertices u and v with the end vertices u ′ and v ′ of the paths P q and P r , respectively. Let G n , d denote the set of simple, connected graphs with n vertices and diameter d . A graph in G n , d that attains the largest spectral radius of the maximum degree matrix is called a maximizing graph. In this paper, we first characterize the spectrum of the maximum degree matrix for graphs of the form B n − i + 2 , i , d − i , where 1 ≤ i ≤ ⌊ d 2 ⌋ . Furthermore, for d ≥ 2 , we prove that the maximizing graph in G n , d is B n − d + 2 , ⌊ d 2 ⌋ , ⌈ d 2 ⌉ . Finally, if d ≥ 4 is an even integer, then the spectral radius of the maximum degree matrix in B n − d + 2 , ⌊ d 2 ⌋ , ⌈ d 2 ⌉ can be computed as the largest eigenvalue of a symmetric tridiagonal matrix of order d 2 + 1 .