Some graft transformations and their applications on distance (signless) Laplacian spectra of graphs

Abstract: Suppose that the vertex set of a connected graph $G$ is $V(G)={v_1,\cdots,v_n}$. 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 $Q_{D}(G)=Tr(G)+D(G)$ and $L_{D}(G)=Tr(G)-D(G)$ are respectively the distance signless Laplacian matrix and distance Laplacian matrix of $G$. The largest eigenvalues $\rho_{Q}(G)$ and $\rho_{L}(G)$ of $Q_D(G)$ and $L_D(G)$ are respectively called distance signless Laplacian spectral radius and distance Laplacian spectral radius of $G$. In this paper we give some graft transformations and use them to characterize the tree $T$ such that $\rho_{Q}(T)$ and $\rho_{L}(T)$ attain the maximum among all trees of order $n$ with given number of pendant vertices.