论文标题
林比率
The Tree-Forest Ratio
论文作者
论文摘要
跨越森林的生根跨度的数量除以带有kirchhoff矩阵k的图G中的生根树的数量是矩阵树和基质森林定理的光谱量tau（g）= det（g）= det（1+k）/det（k）。我们证明，在Barycentric的细化下，树索引t（g）= log（det（k））/n和森林指数f（g）= log（det（det（1+k））/n，因此树孔索引i = f-g = log = log = log（tau（g）/nge（tau（g））/nge to仅取决于该图中最大插入的数字。在1维情况下，所有数字都是已知的：t（g）= 0，f（g）= i（g）= 2 log（phi），其中phi是黄金比率。收敛性证明使用barycentral限制定理，确保Kirchhoff光谱薄弱地收敛到正面真实轴上的度量DK，该轴仅取决于G.树木和森林指数的尺寸i = u（-1）-U（-1）-U（-1）-U（z）= int_r log u（z）= int_r log | x-Z |由riesz度量dk = delta u定义的dk（x），仅取决于g的维度。为所有z的电势u（z）定义为远离dk的支持，而在z = 0处的有限的dk和有限。收敛从尾部估计值K [x，infty] <c exp（-a x）中，其中衰减率a仅取决于最大维度。使用归一化的zeta函数zeta（s）=（1/n）sum_k lambda_k^-s，对于大于1的最大维度的所有有限图，我们都有标识I（g）= sum_t（s+1）^（s+1）zeta（s）zeta（s）/s。限制zeta函数zeta（s）= int_r x^（ - s）dk（x）在s <0中分析。 Hurwitz光谱Zeta Zeta Zeta_z（s）= u_s（z）= int_r（x-z）^（ - s）dk（x）dk（x）补充了u（z）= int_r log（x-z）dk（x-z）dk（x），并且在c- r^+中为z分析z in in c-r^+，并且在c-r^+中是c-r^+ in c。
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.
