Proof of a conjecture on the distance Laplacian spectral radius of graphs.

Let G be a connected graph with vertex set V ( G ) = { v 1 , v 2 , … , v n } and edge set E ( G ) . The distance Laplacian matrix of G is defined as D L ( G ) = T r ( G ) − D ( G ) , where D ( G ) is the distance matrix and T r ( G ) = diag ( t r v 1 , t r v 2 , … , t r v n ) is the diagonal matrix of vertex transmissions of G . The largest eigenvalue of D L ( G ) is called the distance Laplacian spectral radius of G . In this paper, we obtain a graft transformation of a connected graph, which increases its distance Laplacian spectral radius. Using this transformation, we prove a conjecture involving the distance Laplacian spectral radius. [ABSTRACT FROM AUTHOR]