Geodesic
Counting rooted spanning forests in cobordism of two circulant graphs
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 814-823

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We consider a family of graphs $H_n(s_1,\dots,s_k;t_1,\dots,t_\ell),$ which is a generalization of the family of $I$-graphs, which in turn, includes the generalized Petersen graphs and the prism graphs. We present an explicit formula for the number $f_{H}(n)$ of rooted spanning forests in these graphs in terms of Chebyshev polynomials and find its asymptotics. Also, we show that the number of rooted spanning forests can be represented in the form $f_{H}(n)=p a(n)^2,$ where $a(n)$ is an integer sequence and $p$ is a prescribed integer depending on the number of odd elements in the sequence $s_{1},\dots,s_{k},t_{1},\dots,t_{\ell}$ and the parity of $n$.
Keywords: $I$-graph, Petersen graph, prism graph, spanning forest, Chebyshev polynomial, Mahler measure.
Mots-clés : circulant graph 
@article{SEMR_2020_17_a133,
     author = {N. V. Abrosimov and G. A. Baigonakova and L. A. Grunwald and I. A. Mednykh},
     title = {Counting rooted spanning forests in cobordism of two circulant graphs},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {814--823},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a133/}
}
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N. V. Abrosimov; G. A. Baigonakova; L. A. Grunwald; I. A. Mednykh. Counting rooted spanning forests in cobordism of two circulant graphs. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 814-823. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a133/