Extremal eigenvalues of graphs embedded on surfaces

Graph theory on surfaces extends classical graph structures to topological surfaces, providing a theoretical foundation for characterizing the embedding properties of complex networks in constrained spaces. The study of bounding the spectral radius $\rho(G)$ of graphs on surfaces has a rich history that dates back to the 1990s. In this paper, we establish tight bounds for graphs of order $n$ that are embeddable on a surface with Euler genus $\gamma$. Specifically, if graph $G$ achieves the maximum spectral radius, then \begin{equation*} \begin{array}{ll} \frac32\!+\!\sqrt{2n\!-\!\frac{15}4}\!+\!\frac{3\gamma\!-\!1}{n}<\rho(G)<\frac32\!+\!\sqrt{2n\!-\!\frac{15}4}\!+\!\frac{3\gamma\!-\!0.95}{n}, \end{array} \end{equation*} which improves upon the earlier bound $\rho(G)\leq2+\sqrt{2n+8\gamma-6}$ by Ellingham and Zha [JCTB, 2000]. Furthermore, we prove that any extremal graph is obtained from $K_2 \nabla P_{n-2}$ by adding exactly $3\gamma$ edges, where `$\nabla$' means the join product. As a corollary, for $\gamma = 0$ and $n \geq 4.5 \times 10^6$, the graph $K_2 \nabla P_{n-2}$ is the unique planar extremal graph, thereby confirming a long-standing conjecture resolved by Tait and Tobin [JCTB, 2017]. Let $K_r^n$ be the graph of order $n$ obtained by attaching two paths of nearly equal length to two distinct vertices of $K_r$. Integrating spectral techniques with considerable structural analysis on surface graphs, we further derive the following sharp bounds: $\rho(G) \leq \rho(K_2 \nabla K_4^{n-2})$ for projective-planar graphs, and $\rho(G) \leq \rho(K_2 \nabla K_5^{n-2})$ for toroidal graphs. Our study presents a novel framework for exploring the eigenvalue-extremal problem on surface graphs with high Euler genus.