Maxima of the $Q$-index for outer-planar graphs

TL;DR

This paper proves that among all outer-planar graphs of a given size, the graph formed by connecting a single vertex to a path of length n-1 maximizes the signless Laplacian eigenvalue.

Contribution

It establishes the maximum $Q$-index for outer-planar graphs and identifies the extremal graph as $K_1 abla P_{n-1}$.

Findings

The graph $K_1 abla P_{n-1}$ has the largest $Q$-index among outer-planar graphs.

The $Q$-index of this extremal graph is explicitly characterized.

The result provides a spectral extremal characterization for outer-planar graphs.

Abstract

The $Q$-$index$ of graph $G$ is the largest eigenvalue $q(G)$ of its signless Laplacian $Q(G)$. In this paper, we prove that the graph $K_{1}\nabla P_{n-1}$ has the maximal $Q$-index among all outer-planar graphs of order $n$.