arXiv2026
We investigate a classical extremal problem in combinatorics and topological graph theory: determining the maximum spectral radius $spex(n,γ)$ of all $n$-vertex graphs embeddable on a closed surface of Euler genus $γ$. Building on seminal contributions by Boots and Royle, Cao and Vince, and Hong in the 1990s; Ellingham and Zha in 2000; Dvořák and Mohar in 2010; Tait and Tobin in 2017; and Liu, Ning, and Wang in 2026, we provide a complete characterization of spectral extremal graphs for any fixed Euler genus $γ$ and all sufficiently large $n$. The extremal graph attaining $\operatorname{spex}(n,γ)$ is unique, necessarily edge-extremal, and, roughly speaking, is constructed from the join $K_2 \nabla P_{n-2}$ by adding exactly $3γ$ edges. As a consequence, we derive asymptotically tight estimates for $\operatorname{spex}(n,γ)$, strengthening a classical bound due to Ellingham and Zha [J. Combin. Theory B, 2000]. We further characterize the extremal structures for the projective plane and torus, sharpening earlier bounds of Hong [J. Combin. Theory B, 1995]. For the planar case, we establish an explicit vertex threshold of $4.5\times 10^6$ for the Tait--Tobin theorem. Our approach combines extremal constructions of cellular embeddings, topological edge-switching techniques, and walk-counting estimates for large-scale substructures, revealing a bridge between spectral extremality and surface topology.