For a simple graph $G$ on $n$ vertices, let us define

$$\mathcal{I}_{n}(G)=\sum_{i,j=1}^{n}|\mathcal{deg}\ x_{i}-\mathcal{deg}\ x_{j}|^{3}.$$

I know that there are many different topological indices defined and studied for grahs. Have You ever seen such that was defined similar as above? Can You provide any references?

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 - these graphs produce degree sequences that are extreme points of he convex hull of all degree sequences. But this didn't lead me too far. I will be glad for any insight.

For a simple graph $G$ on $n$ vertices, let us define

$$\mathcal{I}_{n}(G)=\sum_{i,j=1}^{n}|\deg\ x_{i}-\deg\ x_{j}|^{3}.$$

I know that there are many different topological indices defined and studied for graphs. Have You ever seen such that was defined similar as above? Can You provide any references?

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 - these graphs produce degree sequences that are extreme points of he convex hull of all degree sequences. But this didn't lead me too far. I will be glad for any insight.