Discrete Analogue of Kokarev's Bound

• The paper introduces a discrete analogue of Kokarev’s bound, establishing extremal inequalities that link graph genus, boundary size, and spectral parameters.
• It employs refined triangulation, circle packing, and branched holomorphic mappings alongside Rayleigh quotient techniques to extend continuous bounds to discrete settings.
• These results have significant applications in spectral geometry, quantum physics, and Boolean function analysis, revealing optimal scaling behaviors across contexts.

The discrete analogue of Kokarev’s bound describes a family of extremal inequalities for spectral, functional, and probabilistic quantities defined on discrete structures (e.g., graphs or cubes) that parallel sharp geometric or analytic inequalities in the continuum. Originally, Kokarev’s bound characterized optimal upper limits on the first (nontrivial) Steklov eigenvalue of a compact Riemannian surface in terms of genus and boundary length. In the discrete context, multiple independent lines of research (graph Steklov problems, discrete Schrödinger operators, and local tail behavior for Boolean polynomials) have established analogues of this phenomenon, demonstrating that the interplay between topology, boundary size, and function growth seen in smooth settings extends to the discrete.

1. Discrete Steklov Eigenvalue Bound for Graphs

For a finite, simple, undirected graph $G = (V,E)$ with a distinguished nonempty boundary subset $\delta\Omega \subset V$ and maximum degree $D$, the first nontrivial Steklov eigenvalue, $\lambda_2(G,\delta\Omega)$, generalizes the spectral Dirichlet-to-Neumann map from Riemannian geometry. The discrete Steklov problem seeks nonconstant $f: V \to \mathbb{R}$ with

$$\Delta f(x) = 0 \quad (x \in V \setminus \delta\Omega), \qquad \partial_\nu f(x) = \lambda f(x) \quad (x \in \delta\Omega),$$

where

$$(\Delta f)(x) = \sum_{\{x,y\}\in E}(f(x) - f(y)), \qquad (\partial_\nu f)(x) = \sum_{\{x,y\}\in E}(f(x) - f(y)).$$

The genus $g$ of $G$ is the smallest $g$ for which $G$ admits an embedding in a closed orientable surface of genus $g$. The discrete analogue of Kokarev's bound, proved in (Chen et al., 19 Nov 2025), states:

For any finite graph $G$ of genus $g$ and bounded degree $D$, with nonempty boundary $\delta\Omega$, $$> \lambda_2(G, \delta\Omega) \leq C(D)\frac{g}{|\delta\Omega|}, >$$ where $C(D)$ depends only on $D$. In particular, for planar graphs ($g=0$), this reduces to $\lambda_2 \leq O(1)/|\delta\Omega|$, recovering previous planar results.

This bound matches the functional dependence of Kokarev's continuous bound, which is $$\lambda_2(M,\partial M) \leq \tfrac{8\pi(g+1)}{|\partial M|}$$ for a compact surface $M$ of genus $g$.

2. Methodology and Proof Techniques

The proof framework in (Chen et al., 19 Nov 2025) leverages a refinement of the topological and analytic structure of $G$:

• Graph Triangulation and Refinement: The graph $G$ is triangulated and further subdivided (hexagonal subdivision) to control the combinatorics, ensuring that the degree and genus are preserved and that boundary size grows in a controlled manner.
• Circle Packing and Riemann Surface Embeddings: The Koebe-Andreev-Thurston theorem provides a circle packing of the refined graph $G^{(k)}$ on a genus-$g$ Riemann surface.
• Branched Holomorphic Mapping: By the Riemann-Roch argument (Kelner), there exists a holomorphic map from the surface to the sphere $S^2$ of degree $O(g)$, with controlled branching.
• Rayleigh Quotient via Vector-Valued Test Functions: The Steklov Rayleigh quotient is evaluated using vector-valued functions derived from pushing boundary circle centers to $S^2$. After Möbius normalization, these vectors have zero mean and unit norm on the boundary.
• Energy Estimates and Summation: The numerator in the Rayleigh quotient corresponds (up to constants) to the total spherical area covered $O(g)$-fold by the map; the denominator is proportional to $|\delta\Omega|$. This yields $$\lambda_2(G,\delta\Omega) \lesssim g/|\delta\Omega|$$.

3. Boundary, Genus, and Scaling Behavior

The upper bound is sharp up to universal constants, both in continuous and discrete settings. The genus $g$ appears linearly in the numerator, while the boundary size $|\delta\Omega|$ controls the denominator. In the case $g=0$, the bound reduces to an $O(1)/|\delta\Omega|$ dependence, consistent with the planar case and corresponding results for trees and block graphs as in (Lin et al., 11 Jul 2024). For graphs with unbounded maximum degree $D$, $C(D)$ grows linearly with $D$; for nonorientable surfaces, analogous bounds are expected to hold.

4. Analogue Results for Discrete Operators and Polynomials

Kokarev-type scaling phenomena also arise in spectral bounds for discrete Schrödinger operators and in hypercontractive inequalities for Boolean polynomials:

• Discrete Birman–Schwinger Bound: For the free discrete Schrödinger operator $H_0$ on $\ell^2(\mathbb{Z}^d)$ and potentials $V$,

$$\|\,|V|^{1/2}(H_0 - z)^{-1}|V|^{1/2}\| \leq C(d)\|V\|_{\ell^{d/3,\infty}}$$

if $d \geq 4$ and $V \in \ell^{d/3,\infty}(\mathbb{Z}^d)$ (Tadano et al., 2018). For uniformly decaying $|V(x)| \leq C(1+|x|)^{-2}$, the bound matches the classical (continuous) result and the critical decay is O($|x|^{-2}$). Nonuniform decays require stronger integrability in the discrete case.

• Local Tail Bounds for Discrete Cube Polynomials: For a degree-$d$ polynomial $P$ on $\{ -1, 1 \}^n$ normalized to mean zero, unit variance, the difference of $e^{-r}$ and $e^{-r-1}$ quantiles satisfies

$a_r - a_{r+1} \leq C d (r+1)^{d/2-1}$

for $r \geq 1$ and universal $C$ (Klartag et al., 2021). This “discrete local Kokarev bound” mirrors the continuous-setting result (where the exponent is $(d-1)/2$), confirming sharp dependence for global tails and postulating optimal order $O(1/r)$ for all $d$.

5. Generalizations and Extensions

The discrete Kokarev-type bounds extend to several contexts:

• Higher Steklov Eigenvalues and Generalized Weights: The circle-packing and flow-deformation arguments permit extension to higher order Steklov eigenvalues $\lambda_k$, as well as edge-weighted and vertex-weighted graphs.
• Nonorientable and Unbounded-Degree Graphs: For graphs embedded in nonorientable surfaces, similar $O(g/|\delta\Omega|)$ scaling is expected. If degree bounds are relaxed, the upper bound incurs multiplicative $D$.
• Block and Tree Structures: The treatment of block graphs and trees refines the boundary vs. structure scaling, showing that modified parameters---block size or leaf number---can replace genus or degree in extremal inequalities for specific graph classes (Lin et al., 11 Jul 2024).

6. Applications, Implications, and Open Questions

Discrete analogues of Kokarev’s bound inform multiple domains:

• Spectral Geometry and Extremal Graph Theory: The genus–boundary interplay underpins sharpness results for extremal graph families, revealing classes where $\lambda_2 \sim g/|\delta\Omega|$ as $g\to\infty$.
• Quantum and Mathematical Physics: Sharp resolvent and Birman–Schwinger bounds underpin unitary equivalence of perturbed lattice operators and enable explicit eigenvalue counts under finite-rank perturbations (Tadano et al., 2018).
• Analysis of Boolean Functions: Local tail inequalities furnish new tools for the concentration of measure on the discrete cube, with direct impact on random graph statistics, isoperimetric inequalities, and higher-order chaos analysis (Klartag et al., 2021).
• Sharpness and Optimality: The continuous setting is known to be asymptotically optimal for large genus. Plausible implication is that the constructed “stringy” graphs in the discrete case also achieve asymptotic sharpness. For local tail gaps of Boolean polynomials, the O($d r^{d/2-1}$) bound is conjectured not to be optimal for $d>2$, with universal $O(1/r)$ seen in explicit examples.

7. Summary Table

Setting	Discrete Bound	Scaling Law
Graph Steklov eigenvalue	$$\lambda_2(G,\delta\Omega) \leq C\, g/|\delta\Omega|$$	Linear in genus, inverse in boundary
Discrete Schrödinger (Birman-Schwinger)	$\|K(z)\| \leq C(d)\|V\|_{\ell^{d/3,\infty}}$ (nonuniform decay)	Uniform in potential norm
Boolean polynomial quantile gaps	$a_r - a_{r+1} \leq C d (r+1)^{d/2-1}$	Degree and quantile-dependent

These results establish a robust “dictionary” between continuous and discrete extremal inequalities, translating sharp geometric-spectral phenomena into analogous combinatorial and analytic statements in the discrete field, with ramifications for spectral geometry, operator theory, and probabilistic combinatorics (Chen et al., 19 Nov 2025, Lin et al., 11 Jul 2024, Tadano et al., 2018, Klartag et al., 2021).