# Complexity of the circulant foliation over a graph

## TL;DR

This paper derives a formula for counting spanning trees in a family of graphs formed by circulant foliations over a base graph, including various well-known graph families, and analyzes their properties and asymptotics.

## Contribution

It provides a closed-form expression for the number of spanning trees in circulant foliation graphs using Chebyshev polynomials, extending understanding of their combinatorial complexity.

## Key findings

- Derived a formula for spanning trees count using Chebyshev polynomials.
- Analyzed arithmetical properties of the spanning trees function.
- Established asymptotic behavior of the number of spanning trees as graph size grows.

## Abstract

In the present paper, we investigate the complexity of infinite family of graphs $H_n=H_n(G_1,\,G_2,\ldots,G_m)$ obtained as a circulant foliation over a graph $H$ on $m$ vertices with fibers $G_{1},\,G_{2},\ldots,G_{m}.$ Each fiber $G_{i}=C_{n}(s_{i,1},\,s_{i,2},\ldots,s_{i,k_{i}})$ of this foliation is the circulant graph on $n$ vertices with jumps $s_{i,1},\,s_{i,2},\ldots,s_{i,k_{i}}.$ This family includes the family of generalized Petersen graphs, $I$-graphs, sandwiches of circulant graphs, discrete torus graphs and others. We obtain a closed formula for the number $\tau(n)$ of spanning trees in $H_{n}$ in terms of Chebyshev polynomials, investigate some arithmetical properties of this function and find its asymptotics as $n\to\infty.$