{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T18:00:11Z","timestamp":1759341611083,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>For a $d$-dimensional cell complex $\\Gamma$ with $\\tilde{H}_{i}(\\Gamma)=0$ for $-1\\leq i &lt; d$, an $i$-dimensional tree is a non-empty collection $B$ of $i$-dimensional cells in $\\Gamma$ such that $\\tilde{H}_{i}(B\\cup \\Gamma^{(i-1)})=0$ and $w(B):= |\\tilde{H}_{i-1}(B\\cup \\Gamma^{(i-1)})|$ is finite, where $\\Gamma^{(i)}$ is the $i$-skeleton of $\\Gamma$. The $i$-th tree-number is defined $k_{i}:=\\sum_{B}w(B)^{2}$, where the sum is over all $i$-dimensional trees. In this paper, we will show that if $\\Gamma$ is acyclic and $k_{i}&gt;0$ for $-1\\leq i \\leq d$, then $k_{i}$ and the combinatorial Laplace operators $\\Delta_{i}$\u00a0 are related by\u00a0 $\\sum_{i=-1}^{d}\\omega_{i}\\,x^{i+1}=(1+x)^{2}\\sum_{i=0}^{d-1}\\kappa_{i} x^{i}$, where $\\omega_{i}=\\log \\det \\Delta_{i}$ and $\\kappa_{i}=\\log k_{i}$.\u00a0 We will discuss various consequences and applications of this equation.<\/jats:p>","DOI":"10.37236\/3403","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:08:36Z","timestamp":1578686916000},"source":"Crossref","is-referenced-by-count":3,"title":["Logarithmic Tree-Numbers for Acyclic Complexes"],"prefix":"10.37236","volume":"21","author":[{"given":"Hyuk","family":"Kim","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Woong","family":"Kook","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2014,3,10]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i1p50\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i1p50\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T06:04:30Z","timestamp":1579241070000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v21i1p50"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,3,10]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2014,1,13]]}},"URL":"https:\/\/doi.org\/10.37236\/3403","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2014,3,10]]},"article-number":"P1.50"}}