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

Dandy Fan; Guoping Wang

paper

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

arXiv:1907.02851

Abstract

Suppose that the vertex set of a connected graph $G$ is $V (G) = {v_{1}, \dots, v_{n}}$ . Then we denote by $T r_{G} (v_{i})$ the sum of distances between $v_{i}$ and all other vertices of $G$ . Let $T r (G)$ be the $n \times n$ diagonal matrix with its $(i, i)$ -entry equal to $T r_{G} (v_{i})$ and $D (G)$ be the distance matrix of $G$ . Then $Q_{D} (G) = T r (G) + D (G)$ and $L_{D} (G) = T r (G) - D (G)$ are respectively the distance signless Laplacian matrix and distance Laplacian matrix of $G$ . The largest eigenvalues $ρ_{Q} (G)$ and $ρ_{L} (G)$ 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 $ρ_{Q} (T)$ and $ρ_{L} (T)$ attain the maximum among all trees of order $n$ with given number of pendant vertices.

10pages