On Jacobian group and complexity of the Y-graph

TL;DR

This paper introduces a straightforward method for calculating the Jacobian group of Y-graphs, explicitly determines the structure for a specific case, and derives a formula for the number of spanning trees using Chebyshev polynomials.

Contribution

It provides a new simple approach for counting Jacobian groups of Y-graphs and explicitly characterizes the structure for the case Y(n; 1, 1, 1).

Findings

Explicit structure of Jacobian group for Y(n; 1, 1, 1).

Closed-form formula for spanning trees using Chebyshev polynomials.

Asymptotic analysis of the number of spanning trees.

Abstract

In the present paper we suggest a simple approach for counting Jacobian group of the $Y$-graph $Y(n; k, l, m).$ In the case $Y(n; 1, 1, 1)$ the structure of the Jacobian group will be find explicitly. Also, we obtain a closed formula for the number of spanning trees of $Y$-graph in terms of Chebyshev polynomials and give its asymtotics.