@article{Tu2023, 
author = {Jianhua Tu and Junyi Xiao and Rongling Lang},
title = {Counting the number of dissociation sets in cubic graphs},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {5},
pages = {10021-10032},
keywords = {extremal graph theory, counting, cubic graphs, dissociation sets, graph polynomials},
url = {https://www.sciopen.com/article/10.3934/math.2023507},
doi = {10.3934/math.2023507},
abstract = {Let    G be a graph. A dissociation set of    G is a subset of vertices that induces a subgraph with vertex degree at most 1. The dissociation polynomial of    G is        D          G        (  λ  )  =      ∑          D      ∈              D            (      G      )            λ                  |            D              |            , where        D    (  G  ) is the set of all dissociation sets of    G. In this paper, we prove that for any cubic graph    G and any    λ  ∈  (  0  ,  1  ],          1                  |            V      (      G      )              |              ln  ⁡      D          G        (  λ  )  ≤      1    4    ln  ⁡      D                  K        4              (  λ  )with equality if and only if    G is a disjoint union of copies of the complete graph        K          4      . When    λ  =  1, the value of        D    G    (  λ  ) is exactly the number of dissociation sets of    G. Hence, for any cubic graph    G on    n vertices,        |        D    (  G  )      |    ≤      |        D    (      K    4    )            |              n              /            4        =      11          n              /            4        .}
}