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

TL;DR

This paper investigates how certain graft transformations affect the spectral radii of the distance signless Laplacian and distance Laplacian matrices of graphs, especially trees, to identify extremal structures.

Contribution

It introduces graft transformations and characterizes trees that maximize the spectral radii of these matrices given the number of pendant vertices.

Findings

Identifies trees with maximum spectral radii for given pendant vertices.

Provides graft transformations to analyze spectral properties.

Characterizes extremal trees for distance Laplacian spectra.

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.