On the spread of outerplanar graphs

Abstract

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 n , the n n -vertex outerplanar graph with maximum spread is a vertex joined to a linear forest with Ω ( n ) \Omega \left(n) edges. We conjecture that the extremal graph is a vertex joined to a path on n − 1 n-1 vertices.