Geodesic
On the number of spanning trees in labeled cactus
Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 139-140

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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, enumeration.
Mots-clés : cactus 
@article{PDMA_2017_10_a53,
     author = {V. A. Voblyi},
     title = {On the number of spanning trees in labeled cactus},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {139--140},
     publisher = {mathdoc},
     number = {10},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2017_10_a53/}
}
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V. A. Voblyi. On the number of spanning trees in labeled cactus. Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 139-140. http://geodesic.mathdoc.fr/item/PDMA_2017_10_a53/