A note on Bartholdi zeta function and graph invariants based on resistance distance

Let G be a finite connected graph. In this note, we show that the complexity of G can be obtained from the partial derivatives at (1−1t,t) of a determinant in terms of the Bartholdi zeta function of G. Moreover, the second order partial derivatives at (1−1t,t) of this determinant can all be expressed as the linear combination of the Kirchhoff index, the additive degree-Kirchhoff index, and the multiplicative degree-Kirchhoff index of the graph G.