The Laplacian spectral radius of graphs on surfaces

Let G be an n  -vertex (n⩾3n⩾3) simple graph embeddable on a surface of Euler genus γγ (the number of crosscaps plus twice the number of handles). Denote by ΔΔ the maximum degree of G. In this paper, we first present two upper bounds on the Laplacian spectral radius of G as follows:(i)λ1(G)⩽Δ+4+(Δ+4)2+8(2n+8γ-10)2.(ii)if G is 4-connected and either the surface is the sphere or the embedding is 4-representative, thenλ1(G)⩽Δ+2+(Δ+2)2+8(2n+2γ-4)2.Some upper bounds on the Laplacian spectral radius of the outerplanar and Halin graphs are also given.