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Bounding the number of arithmetical structures on graphs
Discrete Mathematics ( IF 0.8 ) Pub Date : 2021-06-08 , DOI: 10.1016/j.disc.2021.112494 Christopher Keyes , Tomer Reiter
中文翻译:
限制图上算术结构的数量
更新日期:2021-06-08
Discrete Mathematics ( IF 0.8 ) Pub Date : 2021-06-08 , DOI: 10.1016/j.disc.2021.112494 Christopher Keyes , Tomer Reiter
Let G be a connected undirected graph on n vertices with no loops but possibly multiedges. Given an arithmetical structure on G, we describe a construction which associates to it a graph on vertices and an arithmetical structure on . By iterating this construction, we derive an upper bound for the number of arithmetical structures on G depending only on the number of vertices and edges of G. In the specific case of complete graphs, possibly with multiple edges, we refine and compare our upper bounds to those arising from counting unit fraction representations.
中文翻译:
限制图上算术结构的数量
设G是n个顶点上的无环连通无向图,但可能有多重边。给定算术结构在G 上,我们描述了一个与图相关联的结构 上 顶点和算术结构 上 . 通过迭代这种结构,我们推导的上界算术结构上的数字ģ仅在顶点和的边的数量取决于ģ。在完整图的特定情况下,可能具有多条边,我们将我们的上限与计数单位分数表示产生的上限进行细化和比较。