If ${\cal C}$ is a collection of subsets of a set $X$, we associate to ${\cal C}$ a graph $G\_{\cal C} = (V,E)$ where $V = {\cal C}$ and $$E = \big\{\{A,B\}: A\neq B\in {\cal C} \land A\cap B \neq \emptyset\big\}.$$


If $\kappa$ is a cardinal and ${\cal C}$ is a collection of subsets of $\kappa$, it is easy to see that $G\_{\cal C}$ has at most $\kappa$ connected components. So, suppose that $G = (V,E)$ is a graph with $V = 2^\kappa$ and the number of connected components is at most $\kappa$. Is there a collection ${\cal C}$ of subsets of $\kappa$ such that $G \cong G\_{\cal C}$?



Representability of "large" graphs by "small" intersection graphs
Assuming that $\kappa$ is an **infinite** cardinal, I answer your question with a **characterization.**


For $\mathcal C\subseteq\mathcal P(\kappa),$ the intersection graph $G\_\mathcal C=(V,E)$ has the following properties:  

(1) $|V|\le2^\kappa;$  

(2) there is a collection $\mathcal K$ of cliques, with $|\mathcal K|\le\kappa,$ covering all vertices and edges of the graph; that is, each vertex belongs to an element of $\mathcal K,$ and each pair of adjacent vertices is contained in an element of $\mathcal K.$


**Conversely**, I clain that any graph $G=(V,E)$ satisfying those two conditions is isomorphic to an intersection graph $G\_\mathcal C$ with $\mathcal C\subseteq\mathcal P(\kappa).$


Let $G=(V,E)$ be a graph satisfying (1) and (2). We may assume that $V\subseteq\{X:0\in X\subseteq\kappa\}.$ Let $\mathcal K$ be a collection of cliques in $G,$ of cardinality at most $\kappa,$ covering all vertices and edges of $G.$ Since $|\kappa\times\mathcal K|=\kappa,$ it will suffice to show that $G\cong G\_\mathcal C$ for some $\mathcal C\subseteq\kappa\times\mathcal K.$


For $v\in V$ define $A\_v=\{(\alpha,K)\in\kappa\times\mathcal K:\alpha\in v\in K\}.$


**$v\mapsto A\_v\text{ is injective}:$**  

Suppose $u,v\in V$ and $\alpha\in u\setminus v.$ Choose $K\in\mathcal K$ with $u\in K;$ then $(\alpha,K)\in A\_u\setminus A\_v.$


**$\{u,v\}\in E\implies A\_u\cap A\_v\ne\emptyset$:**  

Choose $K\in\mathcal K$ with $u,v\in\mathcal K;$ then $(0,K)\in A\_u\cap A\_v.$


**For $u\ne v,\ A\_u\cap A\_v\ne\emptyset\implies\{u,v\}\in E:$**  

If $(\alpha,K)\in A\_u\cap A\_v),$ then $u$ and $v$ are both in the clique $K.$


Therefore $G\cong G\_\mathcal C$ where $\mathcal C=\{A\_v:v\in V\}.$


