字典积下的图不变量：香农容量及相关参数
Shannon Capacity and Related Graph Invariants for Lexicographic Products
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中文总结 AI 辅助
该研究分析图字典积下的Lovász θ函数、分数Haemers数和香农容量，证明相关乘性、超乘性性质，确定多类图乘积的香农容量并提出开放问题。
中文摘要 AI 辅助
我们研究有限图的字典积下的Lovász θ函数、分数Haemers数和香农容量。该乘积模型化零误差通信，其中一个图描述类间混淆性，另一个描述每个类内部的混淆性。我们给出三个已知结果的替代、初等且自包含的证明：Lovász θ函数和分数Haemers数在字典积下的乘性，以及分数Lovász θ函数与普通Lovász θ函数相等。我们证明香农容量在两种顺序的字典积下均为超乘性，并将有序积与强积进行比较。结合前两个参数在强积下的乘性，这些比较给出了精确确定字典积容量的界和充分条件。我们表明所得的Lovász和分数Haemers上界不可比，并确定了涉及Kneser图、其补图及Kneser图q-模拟的乘积的香农容量。我们进一步证明完全外因子在字典积下保持香农容量。最后，我们确定了迭代字典积幂的容量，包括自补图的幂（这些自补图是顶点传递或强正则的），并提出了一个关于字典积和强积的开放问题。
英文摘要
This paper studies the Shannon capacity of lexicographic products of finite simple graphs, together with the Lovász theta function and the fractional Haemers number. The Shannon capacity is proved to be supermultiplicative under lexicographic products in either order, and these products are compared with the strong product. We explicitly construct three countably infinite families of lexicographic powers based on the Schläfli graph, the McLaughlin graph, and its second subconstituent; in each family, pairing each member with its complement yields strict supermultiplicativity and arbitrarily large multiplicative gaps. Bounds and exact-capacity criteria for lexicographic products are derived, and the resulting upper bounds are shown to be incomparable. The capacities of lexicographic products involving Kneser graphs, their complements, and $q$-analogues of Kneser graphs are determined. It is also shown that a lexicographic product with a complete outer factor preserves the Shannon capacity of an arbitrary inner factor. The capacities of iterated lexicographic powers are determined, including those of self-complementary graphs that are vertex-transitive or strongly regular. Elementary, self-contained proofs are also given for three known results: the multiplicativity of the Lovász theta function and the fractional Haemers number under lexicographic products, and the equality of the fractional and ordinary Lovász theta functions. Finally, an open problem concerning the Shannon capacities of lexicographic and strong products is posed.
发表机构
- Faculty of Mathematics, Technion–Israel Institute of Technology（以色列理工学院数学学院）
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