On the maximum second eigenvalue of outerplanar graphs

TL;DR

This paper investigates the maximum second eigenvalue of outerplanar graphs, identifying extremal structures for large graphs and extending results to higher eigenvalues and connectivity constraints.

Contribution

It determines the extremal graphs achieving the maximum second eigenvalue for large outerplanar graphs and extends the analysis to higher eigenvalues and connectivity conditions.

Findings

Maximum $oldsymbol{ ext{λ}_2}$ achieved by specific graph constructions for large even n.

Extremal graphs for odd n are characterized by a cut vertex with certain properties.

Asymptotic results for maximum $oldsymbol{ ext{λ}_k}$ among connected outerplanar graphs.

Abstract

For a fixed positive integer $k$ and a graph $G$, let $\lambda_k(G)$ denote the $k$-th largest eigenvalue of the adjacency matrix of $G$. In 2017, Tait and Tobin proved that the maximum $\lambda_1(G)$ 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 $G\setminus \{u\}$ 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 $\lambda_k(G)$ among all connected outerplanar graphs for any fixed $k$.