Kirchhoff index of simplicial networks

Abstract

We introduce a high-dimensional analogue of the Kirchhoff index which is also known as total effective resistance. This analogue, which we call the simplicial Kirchhoff index Kf(X), is defined to be the sum of the simplicial effective resistances of all (d+1)-subsets of the vertex set of a simplicial complex X of dimension d. This definition generalizes the Kirchoff index of a graph G defined as the sum of the effective resistance between all pairs of vertices of G. For a d-dimensional simplicial complex X with n vertices, we give formulas for the simplicial Kirchhoff index in terms of the pseudoinverse of the Laplacian LX in dimension d−1 and its eigenvalues:Kf(X)=n⋅trLX+=n⋅∑λ∈Λ+1λ, where LX+ is the pseudoinverse of LX, and Λ+ is the multi-set of non-zero eigenvalues of LX. Using this formula, we obtain an inequality for a high-dimensional analogue of algebraic connectivity and Kirchhoff index, and propose these quantities as measures of robustness of simplicial complexes. In addition, we derive its integral formula and relate this index to a simplicial dynamical system. We present an open problem for a combinatorial proof of our formula by relating the combinatorial interpretation of Rσ to rooted forests in higher dimensions.