  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 $2$.
