Let $n \in \mathbb{N}$. Let $X = \{x_1, x_2, x_3, \dots, x_n\}$ be a set of distinct positive integers. Let $\mathscr{}(X)$ be the set of all $2$ element subsets of $X$. The values of $x_1, x_2, x_3, \dots, x_n$ are all given to us a priori.
Can we exactly compute $\sum x*y$ taken over all $\{x, y\} \in E$ from the following description of $E$?
$E \subseteq \mathscr{P}(X)$
$|E| = (1/2) * \sum_{x \in X} x $
For all $x$ in $X$, $x = | \{e \in E: x \in e\} |$
For example, if $x = 3$, then there exist distinct $y_1, y_2, y_3$ in $X$ such that
$\{ \{x, y1\}, \{x, y1\}, \{x, y1\} \}$ is the set of all elements of $E$ which contain $x$.
Note that we don't require an explicit description of $E$, only an explicit description of $\sum x*y$
One interpretation is that $X$ is a degree sequence of a graph, where each vertex has unique degree.
$\sum x*y$ is the sum of $\deg(v)*\deg(w)$ such that $\{v, w\}$ is an edge of the graph.
Clearly, the degree sequence is insufficient to determine which pairs of vertices are edges or not. However, we don't care about the edge set directly, we care about the sum of products of degrees.
These are two graphs with the same degree sequence for which the expression $\sum_{(x,y)\ in E} \deg(x) \deg(y)$ is different.
I took it from this answer.