The first three largest values of the spectral norm of oriented bicyclic graphs

Abstract

Let D be a digraph with n vertices and let σ 1 ( D ) , σ 2 ( D ) , . . . , σ n ( D ) be the singular values of the adjacency matrix of D , where σ 1 ( D ) ≥ σ 2 ( D ) ≥ · · · ≥ σ n ( D ) . The spectral norm of D is σ 1 ( D ) . In this paper, we determine the orientations of graphs with the first three largest values of the spectral norm over the family of all orientations of bicyclic graphs with at least 12 vertices.