Asymptotic expansion for cycles in homology classes for graphs

TL;DR

This paper derives an asymptotic expansion with error estimates for counting cycles in homology classes of connected graphs, providing formulas for error coefficients and examples of their calculation.

Contribution

It introduces a detailed asymptotic expansion for cycle counts in homology classes, including explicit formulas for error term coefficients.

Findings

Asymptotic formulas for cycle counts in homology classes

Explicit calculation methods for first error term coefficients

Enhanced understanding of cycle distribution in graph homology

Abstract

In this paper we give an asymptotic expansion including error terms for the number of cycles in homology classes for connected graphs. Mainly, we obtain formulae about the coefficients of error terms which depend on the homology classes and give two examples of how to calculate the coefficient of first error term.