We derive exact values and new bounds for the Shannon capacity of two families of graphs: the $q$-Kneser graphs and the tadpole graphs. We also construct a countably infinite family of connected graphs whose Shannon capacity is not attained by the independence number of any finite strong power. Building on recent work of Schrijver, we establish sufficient conditions under which the Shannon capacity of a polynomial in graphs, formed via disjoint unions and strong products, equals the corresponding polynomial of the individual capacities, thereby reducing the evaluation of such capacities to that of their components. Finally, we prove an inequality relating the Shannon capacities of the strong product of graphs and their disjoint union, which yields alternative proofs of several known bounds as well as new tightness conditions. In addition to contributing to the computation of the Shannon capacity of graphs, this paper is intended to serve as an accessible entry point to those wishing to work in this area.

翻译：本文推导了$q$-Kneser图和蝌蚪图这两类图族香农容量的精确值及新界。我们还构造了一个可数无穷连通图族，其香农容量无法通过任何有限强幂的独立数实现。基于Schrijver近期的工作，我们建立了充分条件，使得通过不相交并集与强积构成的图多项式之香农容量等于各分量容量的对应多项式，从而将此类容量的计算归约为其分量的计算。最后，我们证明了图的强积与不相交并集的香农容量之间的不等式，该不等式不仅为若干已知界提供了替代证明，还导出了新的紧性条件。除推进图香农容量的计算研究外，本文旨在为希望进入该领域的研究者提供易于理解的入门指引。