Let $ G $ be a simple graph. A family of cliques of $ G $ is called an (edge) clique covering for $ G $ if every edge of $ G $ belongs to at least one member of the family. A clique covering in which each edge belongs to exactly one clique, is called a \extit{clique partition}. The minimum size of a clique covering of $G$ is called the \extit{clique cover number} of $G$ and is denoted by $\\cc(G)$. Similarly, the \extit{clique partition number} of $G$, denoted by $ \\cp(G) $, is defined as the minimum number of cliques in a clique partition of $ G $. The subject of clique covering has bee widely studied in recent decades. First time, Erd\\H os et al. in \\cite{enerdos66} presented a close relationship between the clique covering and the set intersection representation. Also, they proved that the clique partition number of a graph on $ n $ vertices cannot exceed ${n^2}/{4} $ (known as Erd\\H os-Goodman-P{\\'o}sa theorem). The connections of clique covering and other combinatorial objects have been explored in the literature (see e.g. \\cite{enWallis87, enRees86}). For a survey of the Also a number of different variants of the clique cover number have been proposed. In this thesis, two clique covering parameters, namely {\extit{edge clique cover sum number}} and {\extit{local clique cover number}} have been studied. The {\\it edge clique cover sum number} (or the {\\it sigma clique cover number}) of a graph $G$, denoted by $\\scc(G)$, is defined as the smallest integer $k$ for which there exists a collection of complete subgraphs of $G$, covering all edges of $G$ such that the sum of cardinality of the cliques is at most $k$. Similarly, the {\\it sigma clique partition number} of $ G $ is defined as the smallest integer $ k $ for which there exists a clique partition of $ G $ such that the sum of sizes of the cliques is at most $k$. By definition, $\\scc(G)\\leq \\scp(G)$. It was conjectured by Katona and Tarj\\' an and proved independently by Chung \\cite{enchung81}, Gy\\H ori and Kostochka \\cite{engyori80} and Kahn \\cite{enkahn81} that every graph on $n$ vertices admits a clique partition such that the sum of number of vertices in these cliques is at most $\\lfloor n^2/2\\rfloor$, i.e. $ \\scp(G)\\leq \\lfloor n^2/2\\rfloor$. This can be considered as a generalization of Erd\\H os-Goodman-P{\\'o}sa theorem. The thesis begins by considering the four well known graph theory concepts, namely, the \extit{set intersection representation}, the \extit{Kneser representation}, the \extit{line hypergraph} and the \extit{clique covering} in Chapter \\ref{chp:chapter1}. The conceptual relations of these concepts and their corresponding parameters are illustrated as well. In Chapter \\ref{chp:chap2}, among some other results, the known bound for $\\scc(G)$ is improved. In particular, it is proved that if $G$ is a graph on $n$ vertices with no isolated vertex and the maximum degree of the complement of $G$ is $ d-1 $, for some integer $d$, then $ \\scc(G)\\leq cnd\\lceil\\log \\left(({n-1})/{(d-1)}\\right)\\rceil$, where $c$ is a constant. Moreover, it is conjectured that this bound is best possible up to a constant factor. Using a well-known result by Bollob{\\'a}s on set systems, it is shown that this conjecture is true for $ d=2 $. Finally, an interpretation of this conjecture as an interesting set system problem is given which can be viewed as a multipartite generalization of Bollob{\\'a}s' two families theorem as follows. Let $d\\geq 2$, $t\\geq 1$ and $\\mathcal{F}= \\{(A^1_i, A^2_i,\\ldots, A^d_i) \\ :\\ 1\\leq i\\leq t \\} $ such that $ A^j_i $ is a set of size $ k_{ij} $ and $ A^j_i \\cap A^{j'}_{i'}=\\emptyset $ if and only if $ i= i' $ and $ j\eq j' $. Then, there exists a function $ f $ and a constant $ c $, such that for every $ t\\geq f(d) $, \\[\\sum_{i,j}{k_{ij}} \\geq c d^2 t\\log t. \\] In Chapter \\ref{chp:chapter3}, we investigate an analogous problem of minimizing the sum of block sizes in a pairwise balanced design, where there are some constraints on the size of one block or the size of the largest block. For every positive integers $n,m$, where $m\\leq n$, let $\\mathscr{S}(n,m)$ be the smallest integer $s$ for which there exists a PBD on $n$ points whose largest block has size $m$ and the sum of its block sizes is equal to $s$. Also, let $\\mathscr{S}'(n,m)$ be the smallest integer $s$ for which there exists a PBD on $n$ points which has a block of size $m$ and the sum of it block sizes is equal to $s$. In this chapter, some lower bounds for $\\mathscr{S}(n,m)$ and $\\mathscr{S}'(n,m)$ are obtained. . <