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 √ 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 ρ(G) of a planar graph G of maximum vertex degree D ≥ 4 satisfies √D ≤ ρ(G) ≤ √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 √ 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 K2,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 ∗Supported in part through a postoctoral position at Simon Fraser University. †On leave from: Institute of Theoretical Informatics, Charles University, Prague, Czech Republic. ‡Supported in part by an NSERC Discovery Grant (Canada), by the Canada Research Chair program, and by the Research Grant P1–0297 of ARRS (Slovenia). §On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia.