图的楔和并的树状度与单纯几何范畴
Arboricity and Simplicial Geometric Category of Wedges and Joins of Graphs
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中文总结 AI 辅助
本文研究图的楔和并运算下树状度的精确公式与上下界，并利用其与单纯几何范畴的关系推导相应公式与估计，最后通过实例计算验证。
中文摘要 AI 辅助
我们研究了树状度在两种基本图运算（即楔和并）下的行为，证明了楔的精确公式，并为并建立了通用的上下界。利用连通图的单纯几何范畴用树状度来刻画的性质，我们推导了单纯几何范畴的楔公式，并获得了图并的相应估计。最后，我们通过构造森林分解及相关的强可坍缩子复形覆盖，对几类图进行了显式计算，以说明这些结果。
英文摘要
We investigate the behavior of arboricity under two fundamental graph operations, namely wedges and joins, proving an exact formula for wedges and establishing general upper and lower bounds for joins. Using the characterization of the simplicial geometric category of connected graphs in terms of arboricity, we derive a wedge formula for simplicial geometric category and obtain corresponding estimates for graph joins. Finally, we illustrate these results through explicit computations for several classes of graphs by constructing forest decompositions and the associated covers by strongly collapsible subcomplexes.
