JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (11): 127-134, 146.doi: 10.6040/j.issn.1671-9352.0.2022.050

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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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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

CLC Number: 

  • O157

Fig.1

Ladder graph Ln"

Fig.2

Corresponding decomposition of Ln"

Fig.3

Indecomposable fragment"

Fig.4

3 matching scenarios of Ln about the associated edges of vertex a2"

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