On the spread of outerplanar graphs

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.