On the maximum spectral radius of planar graphs

TL;DR

This paper studies the upper limits of spectral radius in planar graphs with fixed vertices, establishing bounds and identifying the extremal graph structure for large graphs with dominating vertices.

Contribution

It provides tight bounds on the spectral radius of planar graphs and confirms the extremal structure for graphs with a dominating vertex when the number of vertices is at least 48.

Findings

Maximum spectral radius bounds for planar graphs

Join of P2 and Pn-2 attains maximum spectral radius with dominating vertex

Results valid for graphs with n ≥ 48

Abstract

This paper investigates the maximum spectral radius of planar graphs with concrete fixed number of vertices, providing some tight bounds on the maximum spectral radius of general planar graph resorting to its order, and confirming that among all planar graphs containing dominating vertex with concrete fixed order $n \geq 48$, the join of $P_{2}$ and $P_{n-2}$ attains the maximum spectral radius.