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 Eulerʹs 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/n1/2) of sharpness. We also show that a recent bound on ρ by Hong can be deduced by our method.