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 (2n-1) /3 vertices and (n-2) /3 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 (2n-2) /3 vertices and (n-4) /3 isolated vertices.