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Spectral Determinants of Almost Equilateral Quantum Graphs

Jonathan Harrison, Tracy Weyand

TL;DR

This work investigates a quantum-graph analogue of Kirchhoff's matrix-tree theorem by relating the number of spanning trees to the spectral determinant of the Laplacian on equilateral graphs, then extends to almost-equilateral graphs obtained by perturbing a single edge. Using Friedlander’s Dirichlet-to-Neumann framework, the authors derive explicit first-order perturbations of the spectral determinant for three graph families—complete graphs, complete bipartite graphs, and connected circulants—allowing them to estimate the range of edge-length variations for which the nearest integer to the spectral-determinant-based quantity $T_{\Gamma}$ remains the spanning-tree count. They show that, for these classes, the window of validity is wider than previously proven, indicating a robust connection between the spectral determinant and spanning trees beyond strictly equilateral graphs. The results provide concrete formulas for the perturbed determinants and associated spanning-tree counts, and they highlight the practical potential for applying a quantum-graph version of Kirchhoff’s matrix-tree theorem to a broader set of metric graphs.

Abstract

Kirchoff's matrix tree theorem of 1847 connects the number of spanning trees of a graph to the spectral determinant of the discrete Laplacian [22]. Recently an analogue was obtained for quantum graphs relating the number of spanning trees to the spectral determinant of a Laplacian acting on functions on a metric graph with standard (Neumann-like) vertex conditions [20]. This result holds for quantum graphs where the edge lengths are close together. A quantum graph where the edge lengths are all equal is called equilateral. Here we consider equilateral graphs where we perturb the length of a single edge (almost equilateral graphs). We analyze the spectral determinant of almost equilateral complete graphs, complete bipartite graphs, and circulant graphs. This provides a measure of how fast the spectral determinant changes with respect to changes in an edge length. We apply these results to estimate the width of a window of edge lengths where the connection between the number of spanning trees and the spectral determinant can be observed. The results suggest the connection holds for a much wider window of edge lengths than is required in [20].

Spectral Determinants of Almost Equilateral Quantum Graphs

TL;DR

This work investigates a quantum-graph analogue of Kirchhoff's matrix-tree theorem by relating the number of spanning trees to the spectral determinant of the Laplacian on equilateral graphs, then extends to almost-equilateral graphs obtained by perturbing a single edge. Using Friedlander’s Dirichlet-to-Neumann framework, the authors derive explicit first-order perturbations of the spectral determinant for three graph families—complete graphs, complete bipartite graphs, and connected circulants—allowing them to estimate the range of edge-length variations for which the nearest integer to the spectral-determinant-based quantity remains the spanning-tree count. They show that, for these classes, the window of validity is wider than previously proven, indicating a robust connection between the spectral determinant and spanning trees beyond strictly equilateral graphs. The results provide concrete formulas for the perturbed determinants and associated spanning-tree counts, and they highlight the practical potential for applying a quantum-graph version of Kirchhoff’s matrix-tree theorem to a broader set of metric graphs.

Abstract

Kirchoff's matrix tree theorem of 1847 connects the number of spanning trees of a graph to the spectral determinant of the discrete Laplacian [22]. Recently an analogue was obtained for quantum graphs relating the number of spanning trees to the spectral determinant of a Laplacian acting on functions on a metric graph with standard (Neumann-like) vertex conditions [20]. This result holds for quantum graphs where the edge lengths are close together. A quantum graph where the edge lengths are all equal is called equilateral. Here we consider equilateral graphs where we perturb the length of a single edge (almost equilateral graphs). We analyze the spectral determinant of almost equilateral complete graphs, complete bipartite graphs, and circulant graphs. This provides a measure of how fast the spectral determinant changes with respect to changes in an edge length. We apply these results to estimate the width of a window of edge lengths where the connection between the number of spanning trees and the spectral determinant can be observed. The results suggest the connection holds for a much wider window of edge lengths than is required in [20].

Paper Structure

This paper contains 17 sections, 8 theorems, 82 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Quantum Graphs
  4. Almost Equilateral Edge Transitive Graphs
  5. Almost Equilateral Circulant Graphs
  6. Perturbation Theory
  7. The Matrix $\mathbf{R}$ for a Circulant Graph
  8. Eigenvalues of $\mathbf{R}'$
  9. The Spectral Determinant
  10. Spanning Trees
  11. Complete Graphs and Complete Bipartite Graphs
  12. Circulant Graphs
  13. Summary
  14. Appendix: Eigenvalues of an Almost Equilateral Complete Graph
  15. Appendix: Eigenvalues of an Almost Equilateral Complete Bipartite Graph
  16. ...and 2 more sections

Key Result

Theorem 1

For a connected graph $G$ with $V$ vertices, where $i=1,\dots,V$. The $V\times V$ matrix $\mathbf{L}=\mathbf{D}-\mathbf{A}$ is the discrete Laplacian with $\mathbf{D}$ the diagonal matrix of vertex degrees and $\mathbf{A}$ the adjacency matrix of the graph. The $(V-1)\times (V-1)$ matrix $\mathbf{L}[i]$ is produced by deleting the $i$th row an

Figures (2)

  • Figure 1: The complete bipartite graph $K_{3,5}$.
  • Figure 2: The circulant graph $C_{17}(2,5)$.

Theorems & Definitions (9)

  • Theorem 1: Kirchhoff's Matrix Tree Theorem
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Proposition 5
  • Theorem 6
  • Proposition 7
  • Lemma 8
  • proof