Graphs emerging from finite dimensional vector spaces

Abstract

A vector space over a field is defined as a collection closed under finite vector addition and scalar multiplication. Over the course of time, researchers have delved into exploring the intricate relationships between existing algebraic structures and graphs. This exploration led to the emergence of a distinctive class of graphs derived from vector spaces, following investigations into graphs originating from groups and rings. This thesis undertakes a thorough examination of a well-established algebraic structure known as the non-zero component graph of a finite-dimensional vector space over finite fields. Expanding on this, the thesis introduces the concept of orthogonal component graphs over finitedimensional vector spaces with a particular emphasis on the field Zp. The non-zero component graph of a finite-dimensional vector space over a newlinefinite field is a graph where vertices represent all possible non-zero vectors in newlinethe vector space. Vertices in the graph are made adjacent if they share a common basis vector in their linear combination. The thesis explores a variety of properties relating to distances, domination, and connectivity. Furthermore, it conducts in-depth study of coloring, color connections, topological indices, and centrality-based sensitivity specifically for non-zero component graphs. The concept of orthogonality among vectors in the vector space paves the way for a novel algebraic graph structure the orthogonal component graph. In this graph, vertices represent all possible non-zero vectors in the vector space, and adjacent vertices correspond to orthogonal vectors. The study extends to determining the properties of the orthogonal component graph, particularly in the newlinecontext of the field Z p. Additionally, it characterises the relationship between newlinenon-zero component graphs and orthogonal component graphs. In the latter chapters, the concept of non-zero component signed graphs is introduced and thoroughly discussed.