  A multifamily set representation of a finite simple graph $G$ is a
multifamily $\mathcal{F}$ of sets (not necessarily distinct) for which each set
represents a vertex in $G$ and two sets in $\mathcal{F}$ intersects if and only
if the two corresponding vertices are adjacent. For a graph $G$, an
\textit{edge clique covering} (\textit{edge clique partition}, respectively)
$\mathcal{Q}$ is a set of cliques for which every edge is contained in
\textit{at least} (\textit{exactly}, respectively) one member of $\mathcal{Q}$.
In 1966, P. Erd\"{o}s, A. Goodman, and L. P\'{o}sa (The representation of a
graph by set intersections, \textit{Canadian J. Math.}, \textbf{18},
pp.106-112) pointed out that for a graph there is a one-to-one correspondence
between multifamily set representations $\mathcal{F}$ and clique coverings
$\mathcal{Q}$ for the edge set. Furthermore, for a graph one may similarly have
a one-to-one correspondence between particular multifamily set representations
with intersection size at most one and clique partitions of the edge set. In
1990, S. McGuinness and R. Rees (On the number of distinct minimal clique
partitions and clique covers of a line graph, \textit{Discrete Math.}
\textbf{83} (1990) 49-62.) calculated the number of distinct clique partitions
for line graphs. In this paper, we study the set representations of graphs
corresponding to edge clique partitions in various senses, namely family
representations of \textit{distinct} sets, antichain representations of
\textit{mutually exclusive} sets, and uniform representations of sets with the
\textit{same cardinality}. Among others, we completely determine the number of
distinct family representations and the number of antichain representations of
line graphs.
