Logarithmic Tree-Numbers for Acyclic Complexes
Hyuk Kim, Woong Kook · The Electronic Journal of Combinatorics · 2014
For a $d$-dimensional cell complex $\Gamma$ with $\tilde{H}_{i}(\Gamma)=0$ for $-1\leq i 0$ for $-1\leq i \leq d$, then $k_{i}$ and the combinatorial Laplace operators $\Delta_{i}$ are related by $\sum_{i=-1}^{d}\omega_{i}\,x^{i+1}=(1+x)^{2}\sum_{i=0}^{d-1}\kappa_{i} x^{i}$, where $\omega_{i}=\log \det \Delta_{i}$ and $\kappa_{i}=\log k_{i}$. We will discuss various consequences and applications of this equation.