The Spectral Radius of Graphs on Surfaces

This paper provides new upper bounds on the spectral radius ρ (largest eigenvalue of the adjacency matrix) of graphs embeddable on a given compact surface. Our method is to bound the maximum rowsum in a polynomial of the adjacency matrix, using simple consequences of Euler’s formula. Let γ denote the Euler genus (the number of crosscaps plus twice the number of handles) of a fixed surface Σ. Then (i) for n ≥ 3, every n-vertex graph embeddable on Σ has ρ ≤ 2 +√2n+ 8γ − 6, and (ii) a 4-connected graph with a spherical or 4-representative embedding on Σ has ρ ≤ 1 + √2n+ 2γ − 3. Result (i) is not sharp, as Guiduli and Hayes have recently proved that the maximum value of ρ is 3/2+ √ 2n+ o(1) as n → ∞ for graphs embeddable on a fixed surface. However, (i) is the only known bound that is computable, valid for all n ≥ 3, and asymptotic to √ 2n like the actual maximum value of ρ. Result (ii) is sharp for the sphere or plane (γ = 0), with equality holding if and only if the graph is a ‘double wheel’ 2K1 +Cn−2 (+ denotes join). For other surfaces we show that (ii) is within O(1/n) of sharpness. We also show that a recent bound on ρ by Hong can be deduced by our method. * Supported by NSF Grant Number DMS-9622780