Some properties of the distance Laplacian eigenvalues of a graph

Graph functions maximized on a path

“…The aim of the present note is to give a general approach to problems like the following conjectures of Aouchiche and Hansen [1,2]:…”

“…Given a connected graph G, let D (G) be the distance matrix of G, and let T (G) be the diagonal matrix of the rowsums of D (G). The matrix D L (G) = T (G) − D (G) is called the distance Laplacian of G, and the matrix D Q (G) = T (G) + D (G) is called the distance signless Laplacian of G. The matrices D L (G) and D Q (G) have been introduced by Aouchiche and Hansen and have been intensively studied recently, see, e.g., [1][2][3]5,7,12].…”

“…For general properties of the distance Laplacian and the distance signless Laplacian the reader is referred to [1][2][3].…”

On the distance Laplacian spectral radius of graphs

“…The characteristic polynomial of L(K r ∨ (n − r)K 1 ) has been given in [3] (where K r ∨ (n − r)K 1 is called the complete split graph), from which we have the following result.…”

“…Aouchiche and Hansen [2] showed that the distance Laplacian eigenvalues do not increase when an edge is added, and λ n−1 (G) ≥ n with equality if and only if the complement of G is disconnected. Aouchiche and Hansen [3] showed that for an n-vertex tree T with n ≥ 3, λ 1 (T ) ≥ 2n − 1 with equality if and only if T is the star, and computed the distance Laplacian characteristic polynomials of some graphs. Nath and Paul [7] characterized the n-vertex (connected) graphs G whose complements are trees or unicyclic graphs having λ n−1 (G) = n + 1, and showed that the largest distance Laplacian eigenvalue of a path is simple and the corresponding eigenvector has the similar property like that of a Fiedler vector.…”