Flexibility of eigenvalues for graph Laplacians arising from genus 3 surfaces

Abstract: It is known that the small eigenvalues of the Laplacian of a Riemann surface close to the boundary of the modular space can be well approximated by the eigenvalues of the discrete Laplacian on a certain graph coming from the pair of pants decomposition of the surface. In this paper, we provide a complete description of the sets of eigenvalues of the weighted graph Laplacian for all graphs on four vertices that correspond to a valid pair of pants decomposition of a surface of genus 3.

• The paper presents an explicit algebraic classification of eigenvalue constraints for weighted graph Laplacians derived from pair-of-pants decompositions of genus 3 surfaces.
• It details case-wise analyses for five graph types (star, cycle, path, kite, complete), showing how geometric degeneration influences spectral behavior.
• The study advances spectral theory by addressing an inverse eigenvalue problem, offering practical insights for spectral approximations and computational methods.

Flexibility of Eigenvalues for Graph Laplacians Arising from Genus 3 Surfaces

Introduction and Motivation

The spectral properties of Riemann surfaces, particularly small Laplacian eigenvalues, are central to understanding their geometric and dynamical characteristics. In the context of hyperbolic surfaces, previous works have established lower bounds such as $\lambda_{2g-2}\geq 1/4$ for the $(2g-2)$-th eigenvalue [OR]. The behavior of the first $(2g-3)$ eigenvalues thus becomes especially relevant, both in the study of random surfaces as genus increases [AM1, AM2, HMT] and in degeneration scenarios where eigenvalues approach zero [Colbois]. This paper rigorously describes the flexibility of eigenvalues for weighted graph Laplacians associated with graphs modeling degenerating genus 3 surfaces, specifically via pair-of-pants decompositions. These discrete Laplacians provide accurate approximations for small Laplacian eigenvalues on the surface in the degeneration regime [Colbois].

Surface-to-Graph Correspondence

For a hyperbolic surface of genus $g\geq2$, a partition into pairs of pants by non-intersecting geodesics induces a graph structure whose vertices correspond to connected components of the partition, and whose edges are determined by geodesic connections. Edge weights reflect the geometric data—most notably, the sum of relevant geodesic lengths. The combinatorial Laplacian is defined via a natural quadratic form, and the spectrum of this discrete Laplacian closely tracks the small Laplacian eigenvalues of the underlying Riemann surface, particularly as the surface degenerates. Colbois' theorem [Colbois] ensures that for families of metrics $\{g_t\}$ with shrinking geodesics, the first $2g-3$ Laplacian eigenvalues scale proportionally to the graph Laplacian's spectrum as $t\to 0$.

Main Results: Complete Characterization for Genus 3 Decompositions

The paper provides an explicit, case-wise description of the realizable eigenvalues for graph Laplacians corresponding to all valid pair-of-pants decompositions of genus 3 surfaces. There are five possible graphs on four vertices arising in this way: star ($K_{1,3}$), cycle ($C_4$), path ($P_4$), kite (paw), and complete ($(2g-2)$0) graphs.

Star Graph ($(2g-2)$1)

Eigenvalues $(2g-2)$2 must satisfy $(2g-2)$3 and

$(2g-2)$4

This result is derived via symmetry in the eigenvalue equations and is equivalent to the discriminant condition ensuring three real, non-negative roots for the edge weight cubic equation [FGL]. The strong algebraic structure of this constraint is noteworthy.

Cycle Graph ($(2g-2)$5)

Eigenvalues $(2g-2)$6 require $(2g-2)$7 and $(2g-2)$8. The proof demonstrates that a triple where all values are strictly smaller than half of their sum cannot be realized, but every triple meeting the stated criterion corresponds to a possible weighting.

Path Graph ($(2g-2)$9)

The admissible eigenvalues are characterized by $(2g-3)$0, $(2g-3)$1, and

$(2g-3)$2

where $(2g-3)$3, $(2g-3)$4, $(2g-3)$5. This constraint is obtained by reduction from the cycle graph and careful analysis of root structure.

Kite Graph (Paw)

Allowed eigenvalues $(2g-3)$6 ($(2g-3)$7) satisfy at least one of: $(2g-3)$8 The proof employs parameterization and cubic root structure, yielding sharp boundary conditions.

Complete Graph ($(2g-3)$9)

Any non-negative triple $g\geq2$0 is attainable for $g\geq2$1, confirming that only the complete graph realizes all possible spectra among graphs of four vertices [CdV88, FGL]. The argument uses inductive construction via graph suspension and explicit formulae for Laplacian eigenvalues.

Practical and Theoretical Implications

These results establish explicit spectral flexibility boundaries for degenerating genus 3 surfaces, correlating geometric degeneration with precise spectral behavior. The work clarifies and rigorously resolves instances of the inverse eigenvalue problem for weighted graph Laplacians on these topologies, connecting directly with broader questions in spectral geometry [FGL, CFGL]. Notably, only the complete graph admits all spectra, reflecting profound limitations on attainable eigenvalue sets imposed by graph structure.

For geometric analysis and computational spectral theory, these explicit criteria provide valuable constraints and test cases. In the context of random or highly degenerate surfaces, the discrete Laplacian models become essential for predicting spectral gaps and multiplicities [AM1, AM2, HMT]. The strong algebraic and geometric interplay outlined here raises potential for new algorithms in spectral approximation and for further study of spectral rigidity versus flexibility in both graph and surface settings.

Further Directions and Open Problems

The authors outline open questions regarding the growth of realizable eigenvalue sets as edges are added or suspensions are taken, seeking uniform lower bounds on spectral flexibility expansion and positing conjectures about limits of flexibility for iterated suspensions. The connection to the inverse eigenvalue Laplace problem for general graphs remains active, with recent advances suggesting further generalizations and computational approaches [FGL, CFGL].

Conclusion

This paper offers a comprehensive algebraic classification of possible small Laplacian eigenvalues for graphs modeling degenerating genus 3 surfaces, tightly linking the geometry of surface decompositions with the spectral properties of discrete Laplacians. The results provide explicit spectral flexibility boundaries and exemplify the deep connections between combinatorial graph theory and Riemann surface spectral geometry (2604.26308). The framework herein invites further investigation into spectral geometry, graph Laplacians, and their applications in both mathematical physics and algorithmic spectral analysis.