The Lovász local lemma (LLL) is a central tool in probabilistic combinatorics, providing a sufficient condition under which a finite collection of undesirable events with limited dependencies can be simultaneously avoided with positive probability. This paper offers a self-contained expository treatment of the lemma and its strengthened versions, emphasizing mathematical foundations, conceptual clarity, and applications. We begin with a pedagogically motivated proof of the LLL based entirely on unconditional probability inequalities. Particular attention is given to the symmetric form of the lemma and several subsequent strengthenings. We also discuss a variety of classical applications of both the symmetric and asymmetric forms of the LLL in combinatorics and graph theory, including bounds for the edge-disjoint paths problem, satisfiability of Boolean formulas in conjunctive normal form, lower bounds on diagonal and off-diagonal Ramsey numbers, hypergraph coloring results, structural properties of directed graphs, and acyclic graph colorings. Additional observations and refinements are provided throughout. We also introduce the algorithmic framework of Moser and Tardos, highlighting its constructive counterpart to the LLL, together with an introduction to the entropy-compression principle. The lopsided LLL, a refinement of the LLL, is presented along with an application to the Latin transversal problem. We further discuss the cluster-expansion lemma and its relation to the LLL, and present an alternative treatment of the Latin transversal problem from the cluster-expansion perspective that yields an improved result. The exposition concludes with a high-level overview of the iterated LLL, also known as the semi-random method.

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.

This paper delves into three research directions, leveraging the Lovász $\vartheta$-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 $n\ge 14$, 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 Lovász $\vartheta$-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. (1978). It shows, by a counterexample, that the $\vartheta$-function variant by Schrijver does not possess the property of the Lovász $\vartheta$-function of forming an upper bound on the Shannon capacity of a graph. This research paper also serves as a tutorial of mutual interest in zero-error information theory and algebraic graph theory.