On the maximum spread of planar and outerplanar graphs

TL;DR

This paper determines the extremal graphs with maximum eigenvalue spread among planar and outerplanar graphs, disproving a previous conjecture and identifying specific constructions that maximize spread.

Contribution

It disproves a conjecture by identifying the actual extremal graphs for maximum spread in planar and outerplanar graphs.

Findings

Outerplanar extremal graph joins a vertex to a path and isolated vertices.

Planar extremal graph joins two vertices to a path and isolated vertices.

Disproves previous conjecture on maximum spread in outerplanar graphs.

Abstract

The spread of a graph $G$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $G$. Gotshall, O'Brien and Tait conjectured that for sufficiently large $n$, the $n$-vertex outerplanar graph with maximum spread is the graph obtained by joining a vertex to a path on $n-1$ vertices. In this paper, we disprove this conjecture by showing that the extremal graph is the graph obtained by joining a vertex to a path on $\lceil (2n-1)/3\rceil$ vertices and $\lfloor(n-2)/3\rfloor$ isolated vertices. For planar graphs, we show that the extremal $n$-vertex planar graph attaining the maximum spread is the graph obtained by joining two nonadjacent vertices to a path on $\lceil(2n-2)/3\rceil$ vertices and $\lfloor(n-4)/3\rfloor$ isolated vertices.