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

Abstract 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 − 1 t , t ) of a determinant in terms of the Bartholdi zeta function of G . Moreover, the second order partial derivatives at ( 1 − 1 t , 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 .