The Laplacian spectral radius of graphs on surfaces

Abstract

Let G be an n-vertex ( n ⩾ 3 ) simple graph embeddable on a surface of Euler genus γ (the number of crosscaps plus twice the number of handles). Denote by Δ the maximum degree of G. In this paper, we first present two upper bounds on the Laplacian spectral radius of G as follows: (i) λ 1 ( G ) ⩽ Δ + 4 + ( Δ + 4 ) 2 + 8 ( 2 n + 8 γ - 10 ) 2 . (ii) if G is 4-connected and either the surface is the sphere or the embedding is 4-representative, then λ 1 ( G ) ⩽ Δ + 2 + ( Δ + 2 ) 2 + 8 ( 2 n + 2 γ - 4 ) 2 . Some upper bounds on the Laplacian spectral radius of the outerplanar and Halin graphs are also given.