Spectral radius of finite and infinite planar graphs and of graphs of bounded genus

TL;DR

This paper establishes bounds on the spectral radius of finite and infinite planar graphs and graphs of bounded genus, extending known results for trees and solving an open problem, with applications to hyperbolic tessellations.

Contribution

It derives new bounds for the spectral radius of planar and bounded genus graphs, improves previous results, and introduces a graph decomposition method for hyperbolic tessellations.

Findings

Spectral radius bounds for planar graphs with maximum degree D

Construction of infinite planar graphs reaching the upper bound

Enhanced bounds for hyperbolic tessellations

Abstract

It is well known that the spectral radius of a tree whose maximum degree is $D$ cannot exceed $2\sqrt{D-1}$. In this paper we derive similar bounds for arbitrary planar graphs and for graphs of bounded genus. It is proved that a the spectral radius $\rho(G)$ of a planar graph $G$ of maximum vertex degree $D\ge 4$ satisfies $\sqrt{D}\le \rho(G)\le \sqrt{8D-16}+7.75$. This result is best possible up to the additive constant--we construct an (infinite) planar graph of maximum degree $D$, whose spectral radius is $\sqrt{8D-16}$. This generalizes and improves several previous results and solves an open problem proposed by Tom Hayes. Similar bounds are derived for graphs of bounded genus. For every $k$, these bounds can be improved by excluding $K_{2,k}$ as a subgraph. In particular, the upper bound is strengthened for 5-connected graphs. All our results hold for finite as well as for infinite graphs. At the end we enhance the graph decomposition method introduced in the first part of the paper and apply it to tessellations of the hyperbolic plane. We derive bounds on the spectral radius that are close to the true value, and even in the simplest case of regular tessellations of type $\{p,q\}$ we derive an essential improvement over known results, obtaining exact estimates in the first order term and non-trivial estimates for the second order asymptotics.