On Jacobian group and complexity of I-graph I(n,k,l) through Chebyshev polynomials

TL;DR

This paper introduces a novel method for analyzing the Jacobian group and spanning tree count of I-graphs I(n,k,l), generalizing Petersen graphs, using Chebyshev polynomials to derive formulas and asymptotic properties.

Contribution

It provides a new approach to compute the Jacobian group and spanning trees of I-graphs, including explicit formulas and bounds, extending previous graph theoretical results.

Findings

Minimum number of generators of Jacobian group is between 2 and 2k+2l-1.

Derived a closed-form formula for the number of spanning trees using Chebyshev polynomials.

Investigated arithmetical properties and asymptotic behavior of the spanning tree count.

Abstract

We consider a family of I-graphs I(n,k,l), which is a generalization of the class of generalized Petersen graphs. In the present paper, we provide a new method for counting Jacobian group of the I-graph I(n,k,l). We show that the minimum number of generators of Jac(I(n,k,l)) is at least two and at most 2k + 2l - 1. Also, we obtain a closed formula for the number of spanning trees of I(n,k,l) in terms of Chebyshev polynomials. We investigate some arithmetical properties of this number and its asymptotic behaviour.