The Generating Function is Rational for the Number of Rooted Forests in a Circulant Graph

Siberian Advances in Mathematics Pub Date : 2023-12-14 DOI:10.1134/s1055134423040041

U. P. Kamalov, A. B. Kutbaev, A. D. Mednykh

引用次数: 0

Abstract

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.

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生成函数是圆周图中生根森林数量的有理数

摘要考虑循环图\(\Gamma \)中有根跨越林个数\(f_{\Gamma }(n) \)的生成函数\(\Phi \)，其中\(\Phi (x)= \sum _{n=1}^{\infty } f_{\Gamma }(n) x^n\)和\(\Gamma =C_n(s_1,s_2,\dots ,s_k) \)或\(\Gamma =C_{2n}(s_1,s_2,\dots ,s_k,n) \)。我们证明\(\Phi \)是一个具有整数系数的有理函数，它满足条件\(\Phi (x)=-\Phi (1/x) \)。我们用一系列的例子来说明这个结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Siberian Advances in Mathematics Mathematics-Mathematics (all)

CiteScore

0.70

自引率

0.00%

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期刊介绍： Siberian Advances in Mathematics is a journal that publishes articles on fundamental and applied mathematics. It covers a broad spectrum of subjects: algebra and logic, real and complex analysis, functional analysis, differential equations, mathematical physics, geometry and topology, probability and mathematical statistics, mathematical cybernetics, mathematical economics, mathematical problems of geophysics and tomography, numerical methods, and optimization theory.