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arXiv · 2209.13776

On the maximum spread of planar and outerplanar graphs

Zelong Li·William Linz·Linyuan Lu·Zhiyu Wang
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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.

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Zelong Li, William Linz, Linyuan Lu, Zhiyu Wang. 2022-09-28. On the maximum spread of planar and outerplanar graphs. https://arxiv.org/abs/2209.13776

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