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

Let G be a connected graph with vertex set V(G)={v1,v2,…,vn} and edge set E(G). The distance Laplacian matrix of G is defined as DL(G)=Tr(G)−D(G), where D(G) is the distance matrix and Tr(G)=diag(trv1,trv2,…,trvn) is the diagonal matrix of vertex transmissions of G. The largest eigenvalue of DL(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.