On the maximum second eigenvalue of outerplanar graphs

For a fixed positive integer k and a graph G, let ${\lambda }_k\left(G\right)$ denote the k-th largest eigenvalue of the adjacency matrix of G. In 2017, Tait and Tobin [24] proved that the maximum ${\lambda }_1\left(G\right)$ among all outerplanar graphs on n vertices is achieved by the fan graph $K_1\vee P_{n-1}$. In this paper, we consider a similar problem of determining the maximum ${\lambda }_2$ among all connected outerplanar graphs on n vertices. For n even and sufficiently large, we prove that the maximum ${\lambda }_2$ is uniquely achieved by the graph $(K_1\vee P_{n/2-1})-(K_1\vee P_{n/2-1})$, which is obtained by connecting two disjoint copies of $(K_1\vee P_{n/2-1})$ through a new edge joining their smallest degree vertices. When n is odd and sufficiently large, the extremal graphs are not unique. The extremal graphs are those graphs G that contain a cut vertex u such that $G\setminus \left\{u\right\}$ is isomorphic to $2(K_1\vee P_{n/2-1})$. We also determine the maximum ${\lambda }_2$ among all 2-connected outerplanar graphs and asymptotically determine the maximum of ${\lambda }_k\left(G\right)$ among all connected outerplanar graphs for any fixed k.