Extremal eigenvalues of outerplanar graphs

The extremal eigenvalues including maximum eigenvalues and the minimum eigenvalues about outerplanar graphs are investigated in this paper. Some structural characterizations about the (edge) maximal bipartite outerplanar graphs are represented. With these characterizations, among all bipartite outerplanar graphs of order $n\geq 55$, the maximum spectral radius is completely determined, and moreover, among all general outerplanar graphs of order $n\geq 55$, the minimum least eigenvalue is completely determined.