Advances in the Shannon Capacity of Graphs

This paper studies several research directions concerning the Shannon capacity of graphs. Building on Schrijver's recent framework, we establish sufficient conditions under which the Shannon capacity of a polynomial in graphs equals the corresponding polynomial of the individual capacities, thereby simplifying their evaluation. We derive exact values and new bounds for the Shannon capacity of two families of graphs: the q-Kneser graphs and the Tadpole graphs. Furthermore, we construct graphs whose Shannon capacity is never attained by the independence number of any finite power of these graphs, including a countably infinite family of connected graphs with this property. We further prove an inequality relating the Shannon capacities of the strong product of graphs and their disjoint union, leading to streamlined proofs of known bounds.