On a trace formula of counting Eulerian cycles

TL;DR

This paper establishes a trace formula linking the count of Eulerian cycles in undirected graphs to spectral graph theory, enabling more efficient computation through symmetry considerations.

Contribution

It introduces a novel trace formula connecting Eulerian cycle enumeration to homological spectral graph theory, incorporating symmetry reductions for computational efficiency.

Findings

Derived a trace formula for counting Eulerian cycles

Linked Eulerian cycle counts to spectral properties of graphs

Demonstrated symmetry-based reduction in computation

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.