Spectral Extremal Graphs of Planar Graphs with Fixed Size

TL;DR

This paper determines the spectral extremal graphs among outerplanar and planar graphs with a fixed number of edges, identifying specific structures that maximize spectral radius for large edge counts.

Contribution

It characterizes the spectral extremal graphs for outerplanar and planar graphs with fixed edges, extending previous results to new graph classes and fixed edge counts.

Findings

Maximum spectral radius for outerplanar graphs is achieved by a star when edges are large.

Spectral extremal planar graphs are identified as specific joins of smaller graphs for large edge counts.

Additional extremal graphs for paths, cycles, and complete graphs in these classes are also characterized.

Abstract

Tait and Tobin [J. Combin. Theory Ser. B 126 (2017) 137--161] determined the unique spectral extremal graph over all outerplanar graphs and the unique spectral extremal graph over all planar graphs when the number of vertices is sufficiently large. In this paper we consider the spectral extremal problems of outerplanar graphs and planar graphs with fixed number of edges. We prove that the outerplanar graph on $m \geq 64$ edges with the maximum spectral radius is $S_m$, where $S_m$ is a star with $m$ edges. For planar graphs with $m$ edges, our main result shows that the spectral extremal graph is $K_2 \vee \frac{m-1}{2} K_1$ when $m$ is odd and sufficiently large, and $K_1 \vee (S_{\frac{m-2}{2}} \cup K_1)$ when $m$ is even and sufficiently large. Additionally, we obtain spectral extremal graphs for path, cycle and matching in outerplanar graphs and spectral extremal graphs for path, cycle and complete graph on $4$ vertices in planar graphs.