For a simple graph $G$ on $n$ vertices, let us define
$$\mathcal{I}_{n}(G)=\sum_{i,j=1}^{n}|\operatorname{deg}\ x_{i}-\operatorname{deg}\ x_{j}|^{3}$$
I am highly interested in finding $\sup \mathcal{I}_{n}$ over all graphs with $n$ vertices (or at least some tight upper bound). What I have tried myself, was noticing that $\mathcal{I}_{n}$ must be maximized by a threshold graph:
Sketch of proof: This index $\mathcal{I}_{n}$ is a convex function of degree sequence $\operatorname{deg} \ x_{1},...,\operatorname{deg} \ x_{n}$. Call the set of all such graphic sequences $D$. Then we can look on $D^{∗}=\mathcal{Con}D$ - a convex hull of $D$. It must then attain it's maximum on some extreme point of $D^{∗}$. It can be shown, that such extreme points of $D^{*}$ are exactly those corresponding to threshold graphs.
But this didn't lead me too far. I will be glad for any insight.
Simulations show that the optimum occurs for $k$ isolated vertices and a complete graph on the other $n−k$ where $k=\lfloor\frac{n+1}5\rfloor$. The same count occurs for $k$ vertices of degree $n−1$ and no other edges so the other $n−k$ have degree $k$. Do You have any ideas on how to prove it?