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This paper delves into three research directions, leveraging the Lov\'{a}sz $\vartheta$-function of a graph.
- First, it focuses on the Shannon capacity of graphs, providing new results that determine the capacity for two infinite subclasses of strongly regular graphs, and extending prior results.
- The second part explores cospectral and nonisomorphic graphs, drawing on a work by Berman and Hamud (2024), and it derives related properties of two types of joins of graphs.
- For every even integer such that $n \geq 14$, it is constructively proven that there exist connected, irregular, cospectral, and nonisomorphic graphs on $n$ vertices, being jointly cospectral with respect to their adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, while also sharing identical independence, clique, and chromatic numbers, but being distinguished by their Lov\'{a}sz $\vartheta$-functions.