On the distance Laplacian spectral radius of graphs

Abstract

We determine the unique graphs with minimum distance Laplacian spectral radius among connected graphs with fixed number of pendent vertices, the unique trees with minimum distance Laplacian spectral radius among trees with fixed bipartition, the unique graphs with minimum distance Laplacian spectral radius among graphs with fixed edge connectivity at most half of the number of vertices. We also discuss the minimum distance Laplacian spectral radius of graphs with fixed connectivity. For k = 1, . . . , n−2 2 , we determine the unique n-vertex tree with the (k+1)-th smallest distance Laplacian spectral radius.

The Distance Laplacian Spectral Radius of Clique Trees

“…Recently, Xing and Zhou [3] characterized the unique graph with minimum distance Laplacian spectral radius among all the bicyclic graphs with fixed number of vertices; Aouchiche and Hansen [4] showed that the star K 1,n is the unique tree with the minimum distance Laplacian spectral radius among all trees; Lin et al [5,6] determined the unique graph with minimum distance Laplacian spectral radius among all the trees with fixed bipartition, nonstar-like trees, noncaterpillar trees, nonstar-like noncaterpillar trees, and the graph with fixed edge connectivity at most half of the order, respectively; Niu et al [7] determined the unique graph with minimum distance Laplacian spectral radius among all the bipartite graphs with fixed matching number and fixed vertex connectivity, respectively; Fan et al [8] determined the graph with minimum distance Laplacian spectral radius among all the unicyclic and bicyclic graphs with fixed numbers of vertices, respectively; Lin and Zhou [9] determined the unique graph with maximum distance Laplacian spectral radius among all the unicyclic graphs with fixed numbers of vertices.…”

On distance signless Laplacian spectrum and energy of graphs

“…Studying the eigenvalues of a matrix associated with a graph is the subject of spectral graph theory, where the main objective is determining what characteristics of the graph are reflected in the spectrum of the matrix under consideration. The distance matrix (spectrum) and Laplacian matrix (spectrum) are conceived in full analogy with the ordinary graph energy and their theory is nowadays extensively elaborated (see e.g., [5], [10], [12], [13], [16], [18], [19], [20], [21]). As the distance matrix is very useful in different fields, including the design of communication networks, graph embedding theory as well as molecular stability, therefore maximizing or minimizing the distance spectral radius over a given class of graphs is of great interest and importance.…”

On extremal cacti with respect to the edge Szeged index and edge-vertex Szeged index

“…This topological index has been stuided extensively and has been found applications in modelling physicochemical properties. The upper and lower bounds and other aspects of the Wiener index of many graphs have been fully studied; see, e.g., [1,5,6,8,9,13,19,20,23].…”