Geodesic
The generating function is rational for the number of rooted forests in a circulant graph
Matematičeskie trudy, Tome 26 (2023) no. 2, pp. 129-137 
U. P. Kamalov; A. B. Kutbaev; A. D. Mednykh. The generating function is rational for the number of rooted forests in a circulant graph. Matematičeskie trudy, Tome 26 (2023) no. 2, pp. 129-137. http://geodesic.mathdoc.fr/item/MT_2023_26_2_a5/
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     author = {U. P. Kamalov and A. B. Kutbaev and A. D. Mednykh},
     title = {The generating function is rational for the number of rooted forests in a circulant graph},
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the generating function $\Phi$ for the number $f_\Gamma(n)$ of rooted spanning forests in the circulant graph $\Gamma$, where $\Phi(x)=\sum_{n=1}^\infty f_\Gamma(n)x^n$ and either $\Gamma=C_n(s_1,s_2,\dots,s_k)$ or $\Gamma=C_{2n}(s_1,s_2,\dots,s_k,n)$. We show that $\Phi$ is a rational function with integer coefficients that satisfies the condition $\Phi(x)=-\Phi(1/x)$. We illustrate this result by a series of examples.

[1] F. T. Boesch, H. Prodinger, “Spanning tree formulas and Chebyshev polynomials”, Graphs Combin., 2 (1986), 191–200 | DOI | MR | Zbl

[2] X. B. Chen, “The numbers of spanning trees in undirected circulant graph”, J. Zhang-zhou Teach. Coll. (Nat. Sci.), 13:4 (2000), 1–6 | MR | Zbl

[3] X. B. Chen, Q. Y. Lin, F. J. Zhang, “The number of spanning trees in odd valent circulant graphs”, Discrete Math., 282:1 (2004), 69–79 | DOI | MR | Zbl

[4] L. A. Grunwald, Y. S. Kwon, L. A. Mednykh, “Counting rooted spanning forests for circulant foliation over a graph”, Tokohu Math. J., 74. (2022), 535–548 | MR | Zbl

[5] L. A. Grunwald, L. A. Mednykh, “The number of rooted forests in circulant graphs”, Ars Math. Contemp., 22 (2022), 4.10, 12 pp. | DOI | MR | Zbl

[6] Y. S. Mednykh, A. D. Kwon, I. A. Mednykh, “On Jacobian group and complexity of the generalized Petersen graph $GP(n; k)$ through Chebyshev polynomials”, Linear Algebra Appl., 529 (2017), 355–373 | DOI | MR | Zbl

[7] J. Louis, “Asymptotics for the number of spanning trees in circulant graphs and degenerating d-dimensional discrete tori”, Ann. Comb., 19:3 (2015), 513–543 | DOI | MR | Zbl

[8] J. C. Mason, D. C. Handscomb, Chebyshev Polynomials, Taylor and Francis, London, 2002 | MR

[9] A. D. Mednykh, L.Ȧ. Mednykh, “On the rationality of generating function for the number of spanning trees in circulant graphs”, Algebra Colloq., 27:1 (2020), 87–94 | DOI | MR | Zbl

[10] A. D. Mednykh, L. A. Mednykh, “The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic”, Discrete Math., 342 (2019), 1772–1781 | DOI | MR | Zbl