  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. The third part
focuses on establishing bounds on graph invariants, particularly emphasizing
strongly regular graphs and triangle-free graphs, and compares the tightness of
these bounds to existing ones. The paper derives spectral upper and lower
bounds on the vector and strict vector chromatic numbers of regular graphs,
providing sufficient conditions for the attainability of these bounds. Exact
closed-form expressions for the vector and strict vector chromatic numbers are
derived for all strongly regular graphs and for all graphs that are vertex- and
edge-transitive, demonstrating that these two types of chromatic numbers
coincide for every such graph. This work resolves a query regarding the variant
of the $\vartheta$-function by Schrijver and the identical function by McEliece
et al. It shows, by a counterexample, that the $\vartheta$-function variant by
Schrijver does not possess the property of the Lov\'{a}sz $\vartheta$-function
of forming an upper bound on the Shannon capacity of a graph.
