On the Distance and Distance Laplacian Spectral Radius of Graphs with Cut Edges

Abstract

Suppose that the vertex set of a graph \(G\) is \(V(G) = \{v_1, \ldots, v_n\}\). Then we denote by \({Tr_G}(v_i)\) the sum of distances between \(v_i\) and all other vertices of \(G\). Let \({Tr}(G)\) be the \(n \times n\) diagonal matrix with its \((i,i)\)-entry equal to \({Tr_G}(v_i)\) and \(D(G)\) be the distance matrix of \(G\). Then \(L_p(G) = {Tr}(G) – D(G)\) is the distance Laplacian matrix of \(G\). The largest eigenvalues of \(D(G)\) and \(L_p(G)\) are called distance spectral and distance Laplacian spectral radius of \(G\), respectively. In this paper, we describe the unique graph with maximum distance and distance Laplacian spectral radius among all connected graphs of order \(n\) with given cut edges.