# The effect of a graft transformation on distance signless Laplacian spectral radius of the graphs

## TL;DR

This paper investigates how graft transformations affect the distance signless Laplacian spectral radius of graphs, providing characterizations of extremal trees with respect to this spectral radius.

## Contribution

It introduces graft transformations that influence the spectral radius and characterizes extremal trees among non-starlike and non-caterpillar trees.

## Key findings

- Identifies transformations that minimize or maximize the spectral radius.
- Characterizes graphs with extremal spectral radius among certain tree classes.
- Provides theoretical bounds for the spectral radius based on graph structure.

## 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 $Q_{D}(G)=Tr(G)+D(G)$ is the distance signless Laplacian matrix of $G$. The largest eigenvalues of $Q_D(G)$ is called distance signless Laplacian spectral radius of $G$. In this paper we give some graft transformations on distance signless Laplacian spectral radius of the graphs and use them to characterize the graphs with the minimum and maximal distance signless Laplacian spectral radius among non-starlike and non-caterpillar trees.