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《山东大学学报(理学版)》 ›› 2023, Vol. 58 ›› Issue (11): 127-134, 146.doi: 10.6040/j.issn.1671-9352.0.2022.050

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梯子图双强迫多项式的递推求解

韩慧1,2(),刘雨童2,姚海元1,*()   

  1. 1. 西北师范大学数学与统计学院, 甘肃 兰州 730070
    2. 河北外国语学院, 河北 石家庄 050011
  • 收稿日期:2022-01-05 出版日期:2023-11-20 发布日期:2023-11-07
  • 通讯作者: 姚海元 E-mail:1063064719@qq.com;hyyao@nwnu.edu.cn
  • 作者简介:韩慧(1997—), 女, 硕士研究生, 研究方向为图的匹配理论及其应用. E-mail: 1063064719@qq.com
  • 基金资助:
    国家自然科学基金资助项目(12161081)

Recursive solving of di-forcing polynomials for ladder graphs

Hui HAN1,2(),Yutong LIU2,Haiyuan YAO1,*()   

  1. 1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
    2. Hebei Foreign Studies University, Shijiazhuang 050011, Hebei, China
  • Received:2022-01-05 Online:2023-11-20 Published:2023-11-07
  • Contact: Haiyuan YAO E-mail:1063064719@qq.com;hyyao@nwnu.edu.cn

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摘要：

梯子图Ln是路Pn和路P2的笛卡尔积。图的双强迫多项式是Ln的所有完美匹配的强迫数和反强迫数的二元计数多项式。通过对给定顶点关联边的匹配情况的分类讨论和计数, 得出了梯子图双强迫多项式的递推公式, 并由此计算出了其生成函数和一些低阶梯子图的双强迫多项式。

关键词: 梯子图, 完美匹配, 双强迫多项式, 强迫多项式, 反强迫多项式, 递推关系, 生成函数

Abstract:

The ladder graph Ln is the Cartesian product of the paths Pn and P2. The di-forcing polynomial of Ln is a binary enumerative polynomial of forcing and anti-forcing numbers of all perfect matchings of the graph. We derive a recurrence formula of the di-forcing polynomial for ladder graphsby classification discussion and enumerating of the matching edge associated with a given vertex. And based on this, wecompute the generating function of the di-forcing polynomials for all ladder graphs anddi-forced polynomials for some ladder graphs with low order.

Key words: ladder graph, perfect matching, di-forcing polynomial, forcing polynomial, anti-forcing polynomial, recursion relationship, generating function

中图分类号: 

  • O157

图1

梯子图Ln"

图2

Ln的对应分解"

图3

不可分解片段"

图4

Ln关于顶点a2关联边的3种匹配情况"

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