On the spread of outerplanar graphs

TL;DR

This paper investigates the maximum eigenvalue spread of outerplanar graphs, identifying extremal structures and conjecturing the precise form of the graph that achieves this maximum.

Contribution

It characterizes the structure of outerplanar graphs with maximum spread and proposes a conjecture about the extremal graph configuration.

Findings

Maximum spread is achieved by a vertex joined to a linear forest with many edges.

For large n, the extremal graph has a specific structure involving a vertex and a linear forest.

Conjecture: the extremal graph is a vertex joined to a path on n-1 vertices.

Abstract

The spread of a graph is the difference between the largest and most negative eigenvalue of its adjacency matrix. We show that for sufficiently large $n$, the $n$-vertex outerplanar graph with maximum spread is a vertex joined to a linear forest with $\Omega(n)$ edges. We conjecture that the extremal graph is a vertex joined to a path on $n-1$ vertices.