Eigenvalue-Extremal Problem on Surface Graphs

• The eigenvalue-extremal problem on surface graphs is defined by determining the maximum spectral radius for graphs embeddable on surfaces under Euler genus constraints.
• It integrates algebraic, spectral, and topological techniques to derive precise structural characterizations and sharp eigenvalue bounds.
• The analysis reveals that extremal graphs exhibit a unique join structure with additional edges strategically placed according to topological features.

The eigenvalue-extremal problem on surface graphs concerns the optimal bounds and sharp structural characterizations for extremal spectra of graphs embeddable on topological surfaces, particularly under constraints informed by surface topology such as the Euler genus. This field synthesizes algebraic, spectral, and topological techniques to address how the global geometry of the embedding surface constrains the extremal eigenvalues (usually the spectral radius) of associated graph matrices, identifying not only the extremal values but also the unique combinatorial types of extremal graphs.

1. Definition and Framework

Given an integer $\gamma\geq 0$, let $\mathcal{G}(n, \gamma)$ denote the family of $n$-vertex simple graphs embeddable on a surface of Euler genus $\gamma$. For any $G\in\mathcal{G}(n, \gamma)$, the core focus is the spectral radius $\rho(G)$ of its adjacency matrix. The eigenvalue-extremal problem seeks to determine the maximum possible $\rho(G)$ in $\mathcal{G}(n, \gamma)$ and to characterize all extremal graphs achieving this value.

The spectral extremality is tightly linked to topological invariants due to constraints imposed by graph embeddability. A prototypical instance is the planar case ($\gamma=0$), but the theory generalizes to higher-genus orientable and non-orientable surfaces.

2. Main Results and Structural Theorems

For large enough $n$ in terms of $\gamma$ (specifically, $n\geq 50\cdot(300+180\gamma+24\gamma^2)^2$), the maximizing graph $G$ in $\mathcal{G}(n, \gamma)$ exhibits a rigid structure:

1. $G$ contains $K_2 \nabla P_{n-2}$ (the join of a $2$-clique and a path of length $n-2$) as a spanning subgraph, plus exactly $3\gamma$ additional edges.
2. Writing $\rho_0 = \frac32 + \sqrt{2n-\frac{15}{4}}$, the spectral radius of $G$ satisfies the sharp two-sided estimate:

$$\frac{3}{2} + \sqrt{2n - \frac{15}{4} + \frac{3\gamma-1}{n}} \ <\ \rho(G)\ <\ \frac{3}{2} + \sqrt{2n - \frac{15}{4} + \frac{3\gamma-0.95}{n}}$$

These bounds improve on the classical result of Ellingham–Zha, which gave $\rho(G) \leq 2 + \sqrt{2n + 8\gamma - 6}$, by providing an asymptotically sharper lower-order correction and confirming the sharpness of the $O(1/n)$ term (Zhai et al., 23 Jan 2026).

The extremal configuration is universal: any deviation (such as redistributing the $3\gamma$ additional edges or placing them differently) results in a decrease of the spectral radius due to walk-count monotonicity under edge-switching.

3. Proof Techniques and Methodological Advances

The proof combines combinatorial topology, spectral graph theory, and careful walk-count analysis:

• Edge and Face Counting via Euler’s Formula: For a graph embeddable on a surface of Euler genus $\gamma$, the edge count is bounded by $e(G)\leq 3(n-2+\gamma)$ due to the triangulation face structure, which imposes a maximum average degree.
• Spectral Partition via Perron–Frobenius Theory: Partitioning the graph by entries in the Perron eigenvector isolates "large-entry" vertices; detailed degree-sum and neighborhood arguments force two almost-universal vertices, confirming the $K_2\nabla P_{n-2}$ join.
• Embedding-Face Interlacing: Minimal-embedding ("cellular") combinatorics ensure the cyclic links around high-degree vertices are forced, and additional local face constraints determine the exact placement of the extra $3\gamma$ edges.
• Walk-Count Interlacing: Bounds on the number of 2-walks and 3-walks (closed walks of given lengths) are employed to compare spectral radii under edge modifications, tying spectral increments directly to structural changes.

4. Classification for Low Genus: Planar, Projective-Planar, and Toroidal Extremals

Surface Type	Euler Genus $\gamma$	Extremal Graph Structure	Spectral Radius Bound
Planar	0	$K_2 \nabla P_{n-2}$	$\sim 3/2 + \sqrt{2n - 15/4}$
Projective-Planar	1	$K_2 \nabla K_4^{n-2}$	$\rho(G) \leq \rho(K_2 \nabla K_4^{n-2})$
Toroidal	2	$K_2 \nabla K_5^{n-2}$	$\rho(G) \leq \rho(K_2 \nabla K_5^{n-2})$

For planar graphs and sufficiently large $n$ (e.g., $n\geq 4.5\times 10^6$), $K_2 \nabla P_{n-2}$ is the unique maximizer, resolving the conjecture of Boots–Royle–Cao–Vince and matching the range established by Tait–Tobin. In the projective-planar and toroidal cases, the unique extremal structures are explicit joins with path-attached cliques of suitable order ($K_4$ or $K_5$) (Zhai et al., 23 Jan 2026).

5. Generalizations, Related Problems, and Context

The eigenvalue-extremal problem situates itself at the intersection of graph minor theory, spectral extremal combinatorics, and structural graph theory on surfaces. It complements classical questions about extremal subgraph density and expands the arsenal of spectral tools for minor-closed graph classes (those forbidding certain minors).

This advances earlier work by Mohar, Hong, Ellingham–Zha, and others who developed spectral-radius bounds for surface-embedded graphs, typically relying on global density controls and not achieving the sharp lower-order corrections. The methods apply equally to higher-genus surfaces or minor-closed classes with bounded edge density and are likely extensible to extremal problems for other eigenvalues or Laplacian variants.

6. Technical Innovations and Impact

Key methodological innovations include:

• Iterative Embedding Constructions: Explicit genus-increment constructions via fixed-size gadgets preserve extremality across inductively increasing genera.
• Eigenvector Partition–Combinatorial Synthesis: Partitioning from the spectral side and matching to topological constraints produces tight control over where structural complexity must be concentrated.
• Walk-Count Interlacing: By relating changes in the spectral radius to local walk counts (up to lengths $\Omega(n)$), the analysis translates spectral increases/decreases under edge alterations into combinatorial terms.
• Sharp Correction Terms: The main spectral radius bound’s $O(1/n)$ correction is shown to be optimal, enabled by fine-grained degree-face-embedding analysis unavailable in earlier works.

This framework settles the extremal spectral problem for large $n$ on all closed surfaces of fixed Euler genus, classifies all extremal graphs, gives nearly exact formulae for $\rho(G)$, and lays the groundwork for a unified spectral-extremal theory for surface-embeddable graphs (Zhai et al., 23 Jan 2026).