Maximum principal ratio of the signless Laplacian of graphs

TL;DR

This paper investigates the maximum principal ratio of the signless Laplacian among all connected graphs of large order, identifying the extremal graph as a specific kite graph structure.

Contribution

It determines the extremal connected graph that maximizes the principal ratio of the signless Laplacian for large graphs.

Findings

Extremal graph is a kite graph formed by attaching a path to a complete graph.

Maximum principal ratio is achieved by the kite graph for sufficiently large n.

Provides bounds and characterization of the principal ratio in relation to graph structure.

Abstract

Let $G$ be a connected graph and $Q(G)$ be the signless Laplacian of $G$. The principal ratio $\gamma(G)$ of $Q(G)$ is the ratio of the maximum and minimum entries of the Perron vector of $Q(G)$. In this paper, we consider the maximum principal ratio $\gamma(G)$ among all connected graphs of order $n$, and show that for sufficiently large $n$ the extremal graph is a kite graph obtained by identifying an end vertex of a path to any vertex of a complete graph.