On the maximum spectral radius of planar graphs

TL;DR

This work addresses the problem of determining the maximum spectral radius among planar graphs with a fixed order, grounding the discussion in the join structure $P_{2}$ and $P_{n-2}$. It develops tight, order-dependent bounds leveraging the Rayleigh quotient and structural properties of planar graphs, and shows that among graphs with a dominating vertex for $n \ge 48$, the extremal graph is the join $P_{2}\\nabla P_{n-2}$. The results sharpen existing asymptotic insights by providing explicit lower bounds in several $n$-ranges and establish the exact extremal form within the dominating-vertex family, contributing to the broader extremal spectral theory of planar graphs.

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.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3