Effective Resistances, Kirchhoff index and Admissible Invariants of Ladder Graphs

TL;DR

This paper derives explicit formulas for effective resistances, Kirchhoff index, and invariants of ladder graphs using circuit reduction, trigonometric sums, and Chebyshev polynomials, advancing the analytical understanding of these graph invariants.

Contribution

It provides explicit formulas for effective resistances, Kirchhoff index, and invariants of ladder graphs, connecting them with trigonometric functions and Chebyshev polynomials.

Findings

Explicit formulas for effective resistances between any two vertices.

Sum formula for Kirchhoff index involving trigonometric functions.

Representation of formulas using generalized Fibonacci numbers.

Abstract

We explicitly compute the effective resistances between any two vertices of a ladder graph by using circuit reductions. Using our findings, we obtain explicit formulas for Kirchhoff index and admissible invariants of a ladder graph considering it as a model of a metrized graph. Comparing our formula for Kirchhoff index and previous results in literature, we obtain an explicit sum formula involving trigonometric functions. We also expressed our formulas in terms of certain generalized Fibonacci numbers that are the values of the Chebyshev polynomials of the second kind at $2$.