The Tree-Forest Ratio

TL;DR

This paper investigates the spectral properties of graphs through the tree-forest ratio, demonstrating convergence of certain indices under Barycentric refinements and relating them to potential theory and zeta functions.

Contribution

It introduces the convergence of tree and forest indices to dimension-dependent constants and connects spectral measures to potential theory and zeta functions in graph theory.

Findings

Tree and forest indices converge to dimension-dependent constants.

Spectral measures converge weakly to a measure depending on the graph's dimension.

The paper establishes identities linking the tree-forest index to spectral zeta functions.

Abstract

The number of rooted spanning forests divided by the number of spanning rooted trees in a graph G with Kirchhoff matrix K is the spectral quantity tau(G)= det(1+K)/det(K) of G by the matrix tree and matrix forest theorems. We prove that that under Barycentric refinements, the tree index T(G)=log(det(K))/n and forest index F(G)=log(det(1+K))/n and so the tree-forest index i=F-G=log(tau(G))/n converge to numbers that only depend on the size of the maximal clique in the graph. In the 1-dimensional case, all numbers are known: T(G)=0, F(G)=i(G) =2 log(phi), where phi is the golden ratio. The convergent proof uses the Barycentral limit theorem assuring the Kirchhoff spectrum converges weakly to a measure dk on the positive real axis that only depends on dimension of G. Trees and forests indices are potential values i = U(-1)-U(0) for the subharmonic function U(z)=int_R log|x-z| dk(x) defined by the Riesz measure dk=Delta U which only depends on the dimension of G. The potential U(z) is defined for all z away from the support of dk and finite at z=0. Convergence follows from the tail estimate k[x,infty] < C exp(-a x) where the decay rate a only depends on the maximal dimension. With the normalized zeta function zeta(s) = (1/n) sum_k lambda_k^-s, we have for all finite graphs of maximal dimension larger than 1 the identity i(G) = sum_t (-1)^(s+1) zeta(s)/s. The limiting zeta function zeta(s) = int_R x^(-s) dk(x) is analytic in s for s<0. The Hurwitz spectral zeta function zeta_z(s)=U_s(z) = int_R (x-z)^(-s) dk(x) complements U(z) = int_R log(x-z) dk(x) and is analytic for z in C - R^+ and for fixed z in C-R^+ is an entire function in s in C.