Executive Summary : | The study of graphs from algebraic structures is crucial for understanding the characterization of various algebraic structures and their applications in real-world applications. Graphs are data structures used to describe relationships with entities and can be constructed in various ways. The research aims to find the theoretical properties of graphs associated with vector spaces and apply them to real-world applications. Vector spaces are used to solve linear equations and are often unexpected spaces. Research has been conducted on associating graph structures with various algebraic structures, such as semigroups, groups, rings, modules, and vector spaces. Recent studies have explored intersection graphs, subspace inclusion graphs, nonzero component graphs, and non-zero component union graphs. Graph-theoretic properties such as connectivity, girth, diameter, clique number, Eulerian, Hamiltonian, planar, toroidal, and outer planar are considered. Graph embedding is found in various real-world scenarios, such as communication networks, electric networks, social media networks, and biological networks. The proposed area of research focuses on the properties of graphs constructed from vector spaces, relating diameter, girth, chromatic number, clique number, and domination number. The outcome of this work will be applied to proteins in biological interaction networks and predict new therapeutic applications of existing drug molecules, whose structure can be represented as a graph. |