The idea of “distinguishing the vertices of a graph from one another” goes back to the work by Entringer and Gassman [36] and Sumner [81], where the authors posed a problem: which graphs have property that “there is one-to-one correspondence between vertices and their neighbourhoods”. The vertices of such graphs can be distinguished by their neighbourhoods. The idea has demonstrated its fundamental nature through a wide variety of applications associated to graphs in theory of networks, communication, robot navigation, programming a robot in manipulating objects to name a few. Later work on distinguishability parameters of graphs has used ideas different from the work of Sumner. The following approaches to the problem have found more attention than others. In the distance-based approach, each vertex of a connected graph Γ is distinguished from every other vertex of Γ by labeling a subset M of V (Γ) and using the distances between the vertices of Γ and those of M to construct a one-to-one function on V (Γ). The minimum cardinality of set M is called the metric dimension of Γ. In the symmetry breaking approach, we choose a set of vertices of Γ which has only the trivial automorphism in its stabilizer (only the trivial automorphism fixes the vertices in S). Another idea in symmetry breaking approach is that we choose a set D of vertices and color them with the minimum number of colors. In both of the ideas, the automorphism group of graph Γ is destroyed and we are assured that every vertex of graph can be distinguished. The minimum cardinality of set S is called the fixing number of Γ and the minimum number of colors required to color the vertices of set D is called the distinguishing number of Γ. In the location-domination approach, we choose a dominating set L of a graph Γ such that every vertex of Γ outside the dominating set is uniquely distinguished by its neighborhood within the dominating set. In the covering code approach, we find a set of vertices whose neighborhoods uniquely overlap at any vertex of graph Γ. The brief details of our contributions to this area are as follows: We define a new distinguishability parameter ‘fixed number’ of graphs that gives us the minimum number of vertices with random choice such that fixing those vertices destroys the automorphism group of the graph. We extend the study of weak total resolving set, weak total metric dimension and weak total resolving number of graphs. We also study well-known distinguishingability parameters locating-dominating sets for functigraphs and locating-dominating sets, identifying codes and distinguishing number for non-zero component graphs associated to finite vector space. A set of vertices S of a graph Γ is called a fixing set of Γ, if only the trivial automorphism of Γ fixes every vertex in S. The fixing number of a graph is the smallest cardinality of a fixing set. The fixed number of a graph Γ is the minimum number γ, such that every subset of vertices of Γ with cardinality γ is a fixing set of Γ. A graph Γ is called a γ-fixed graph, if its fixing number and fixed number are both γ. We study the fixed number of a graph and give a construction of a graph of higher fixed number from a graph of lower fixed number. We find bound on γ in terms of the diameter of a distance-transitive γ-fixed graph. A resolving set of vertices M ⊆ V (Γ) is called a weak total resolving set of Γ, if for each vertex σ ∈ M (symbols σ, ρ are used to represent the vertices of a graph) and for each ρ ∈ V (Γ) \ M, there is one element in M \ {σ} that resolves σ and ρ. The smallest cardinality of a weak total resolving set is called the weak total metric dimension of Γ. In this thesis, we extend the study of weak total resolving sets. We give some characterization and realization results on weak total metric dimension and weak total resolving number. We find weak total metric dimension of tree graph. We also define randomly weak total γ-dimensional graph and study its properties. We find weak total resolving sets and weak total metric dimension of functigraphs of some families of graphs. A subset L of the vertices of a graph Γ is called a locating-dominating set of Γ if for every two distinct vertices σ, ρ ∈ V (Γ) \ L, we have ∅ 6= NΓ(σ) ∩ L 6= NΓ(ρ) ∩ L 6= ∅. The location-domination number of Γ is the minimum cardinality of a locating-dominating set in Γ. Let Γ1 and Γ2 be the disjoint copies of a graph Γ and η : V (Γ1) → V (Γ2) be a function. A functigraph FΓ η consists of vertex set V (Γ1) ∪ V (Γ2) and edge set E(Γ1) ∪ E(Γ2) ∪ {σρ : ρ = η(σ)}. We study the variation of location-domination number in passing from Γ to FΓ η and find its sharp lower and upper bounds. We also study the location-domination number of functigraphs of complete graphs for all possible definitions of function η. We also obtain the location-domination number of functigraphs of a family of spanning subgraphs of complete graphs. We investigate the problem of covering the vertices of non-zero component graphs associated to finite vector spaces as introduced by Das [33], such that we can uniquely identify any vertex by examining the vertices that cover it. We use locating-dominating sets and identifying codes, which are closely related concepts for this purpose. We find the location-domination number and the identifying number of the graph and study the exchange property for locatingdominating sets and identifying codes. We extend the study of properties of automorphisms of non-zero component graphs associated to finite vector spaces. We prove that the symmetric group of basis vectors is isomorphic to the automorphism group of the graph. We find the distinguishing number of the graph for both of the cases, when number of field elements are 2 and more than 2.