On the maximum spread of planar and outerplanar graphs

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 $\lceil (2n-1)\mathrm{/}3\rceil$ vertices and $\lfloor (n-2)\mathrm{/}3\rfloor$ 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 $\lceil (2n-2)\mathrm{/}3\rceil$ vertices and $\lfloor (n-4)\mathrm{/}3\rfloor$ isolated vertices.