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On a trace formula of counting Eulerian cycles

Ye Luo

TL;DR

This work connects counting Eulerian circuits on undirected graphs to homological spectral graph theory by deriving an explicit trace formula that expresses ec(G) as a normalized sum of traces of powers of twisted adjacency matrices. The method uses a spanning tree to realize the first homology group via a 2-torsion module and aggregates contributions from twisted vertex and edge adjacencies, controlled by signs determined by a parity function σ(γ). It also provides a directed analogue for Eulerian orientations and demonstrates computational reductions from spectral antisymmetry and automorphisms. The results yield exact counts in examples and offer practical reduction strategies for large graphs, highlighting the interplay between topology, spectral theory, and combinatorial graph counting.

Abstract

We make connections of a counting problem of Eulerian cycles for undirected graphs to homological spectral graph theory, and formulate explicitly a trace formula that identifies the number of Eulerian circuits on an Eulerian graph with the trace sum of certain twisted vertex and edge adjacency matrices of the graph. Moreover, we show that reduction of computation can be achieved by taking into account symmetries related to twisted adjacency matrices induced by spectral antisymmetry and graph automorphisms.

On a trace formula of counting Eulerian cycles

TL;DR

This work connects counting Eulerian circuits on undirected graphs to homological spectral graph theory by deriving an explicit trace formula that expresses ec(G) as a normalized sum of traces of powers of twisted adjacency matrices. The method uses a spanning tree to realize the first homology group via a 2-torsion module and aggregates contributions from twisted vertex and edge adjacencies, controlled by signs determined by a parity function σ(γ). It also provides a directed analogue for Eulerian orientations and demonstrates computational reductions from spectral antisymmetry and automorphisms. The results yield exact counts in examples and offer practical reduction strategies for large graphs, highlighting the interplay between topology, spectral theory, and combinatorial graph counting.

Abstract

We make connections of a counting problem of Eulerian cycles for undirected graphs to homological spectral graph theory, and formulate explicitly a trace formula that identifies the number of Eulerian circuits on an Eulerian graph with the trace sum of certain twisted vertex and edge adjacency matrices of the graph. Moreover, we show that reduction of computation can be achieved by taking into account symmetries related to twisted adjacency matrices induced by spectral antisymmetry and graph automorphisms.

Paper Structure

This paper contains 11 sections, 22 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homology groups of graphs
  4. Twisted vertex and edge adjacency matrices of graphs
  5. BEST theorem
  6. Counting circuits in homology classes
  7. Proof of the main theorem and its directed version
  8. Examples
  9. Reduction of computation
  10. Reduction based on spectral antisymmetry of twisted graph adjacency
  11. Reduction based on graph automorphisms

Key Result

Theorem 1.1

Let $G$ be an Eulerian graph of genus $g$ with $m$ edges, and $T$ a spanning tree of $G$. Then we have the following trace formula for the number of Eulerian cycles on $G$:

Figures (1)

  • Figure 1: (a) A non-Eulerian graph $G_1$; (b) An Eulerian graph $G_2$

Theorems & Definitions (55)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5: BEST theorem
  • Definition 3.1
  • Lemma 3.2
  • Theorem 3.3
  • ...and 45 more