Prikladnaya Diskretnaya Matematika. Supplement
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Prikladnaya Diskretnaya Matematika. Supplement, 2017, Issue 10, Pages 139–140
DOI: https://doi.org/10.17223/2226308X/10/54
(Mi pdma366)
 

Applied Theory of Coding, Automata and Graphs

On the number of spanning trees in labeled cactus

V. A. Voblyi

Bauman Moscow State Technical University, Moscow
References:
Abstract: Let $t(Ca_n(n_2,n_3,\ldots))$ be a number of spanning trees in a labelled cactus with $n$ vertices, $n_2$ be a number of its edge-blocks, $n_2\ge0$, $n_i$ be a number of its polygon-blocks with $i$ vertices, $n_i\ge0$, $i\ge3$, and $k$ be a number of cycles in this cactus. We deduce the formula $t(Ca_n(n_2,n_3,\dots))=\prod_{i\ge3}i^{n_i}$, $n\ge2$, where $n-1=n_2+2n_3+\dots$ As consequence, we obtain inequalities $t(Ca_n(n_2,n_3,\dots))\le(\frac1k(n+k-n_2-1))^k\le(\frac1k(n+k-1))^k\le e^{n-1}$.
Keywords: spanning tree, cactus, enumeration.
Document Type: Article
UDC: 519.175.3
Language: Russian
Citation: V. A. Voblyi, “On the number of spanning trees in labeled cactus”, Prikl. Diskr. Mat. Suppl., 2017, no. 10, 139–140
Citation in format AMSBIB
\Bibitem{Vob17}
\by V.~A.~Voblyi
\paper On the number of spanning trees in labeled cactus
\jour Prikl. Diskr. Mat. Suppl.
\yr 2017
\issue 10
\pages 139--140
\mathnet{http://mi.mathnet.ru/pdma366}
\crossref{https://doi.org/10.17223/2226308X/10/54}
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