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

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

The planar Cayley graphs are effectively enumerable I: consistently planar graphs

We obtain an effective enumeration of the family of finitely generated groups admitting a faithful, properly discontinuous action on some 2-manifold contained in the sphere. This is achieved by introducing a type of group presentation capturing exactly these groups. Extending this in a companion paper, we find group presentations capturing the planar finitely generated Cayley graphs. Thus we obtain an effective enumeration of these Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al.Comment: To appear in Combinatorica. The second half of the previous version is arXiv:1901.0034

On planar Cayley graphs and Kleinian groups

Let $G$ be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface $X \subseteq \mathbb{S}^2$. We prove that $G$ admits such an action that is in addition co-compact, provided we can replace $X$ by another surface $Y \subseteq \mathbb{S}^2$. We also prove that if a group $H$ has a finitely generated Cayley (multi-)graph $C$ covariantly embeddable in $\mathbb{S}^2$, then $C$ can be chosen so as to have no infinite path on the boundary of a face. The proofs of these facts are intertwined, and the classes of groups they define coincide. In the orientation-preserving case they are exactly the (isomorphism types of) finitely generated Kleinian function groups. We construct a finitely generated planar Cayley graph whose group is not in this class. In passing, we observe that the Freudenthal compactification of every planar surface is homeomorphic to the sphere

Behavior of Petrie Lines in Certain Edge-Transitive Graphs

We survey the construction and classification of one-, two- and infinitely-ended members of a class of highly symmetric, highly connected infinite graphs. In addition, we pose a conjecture concerning the relationship between the Petrie lines and ends of some infinitely-ended members of this class