Maxima of the signless Laplacian spectral radius for planar graphs

TL;DR

This paper identifies the planar graph with the maximum signless Laplacian spectral radius for large graphs, specifically proving that the join of K2 and a path maximizes this spectral property.

Contribution

It establishes a extremal spectral graph theory result for planar graphs, pinpointing the specific structure that maximizes the signless Laplacian spectral radius.

Findings

K_{2} abla P_{n-2} has the maximal spectral radius among planar graphs for n ≥ 456.

The result applies to large graphs, providing a clear extremal example.

The proof advances understanding of spectral properties in planar graph classes.

Abstract

The signless Laplacian spectral radius of a graph is the largest eigenvalue of its signless Laplacian. In this paper, we prove that the graph $K_{2}\nabla P_{n-2}$ has the maximal signless Laplacian spectral radius among all planar graphs of order $n\geq 456$.