I have a problem about graph theory and I need some hints to solve it.
Consider a graph $G$ with the vertex set $V(G)$ equal to the set of all two element subsets of the set $A=\{1, 2, 3, 4, 5\}$. Further let an edge $XY$ be in $E(G)$ whenever the two-element subsets $X$ and $Y$ are not disjoint and neither are they identical ($X\cap Y \ne\varnothing\land X\ne Y$). Draw the graph $G$. Determine if $G$ is Hamiltonian and justify your answer.
There are 25 two-element subsets of $A$. So I created a complete graph with 25 vertices. But it seems ridiculous. Also I couldn't get what the problem wanted from me and how can I do that.
What made you think there were 25 two-element subsets of $A$? There are only 10, and $G$ is shown below.
$G$ is Hamiltonian, as evidenced by the outer cycle of edges:
$$12-23-35-54-41-15-52-24-43-31$$