# Volume of the convex hull of the set of all graphic sequences of a given length

Consider the set of all graphic sequences with $n$ elements as a subset of $\mathbb{R}^{n}$, namely let $$D(n)=\{(d_{1},\dots,d_{n})\in\mathbb{Z}_{+}^{n}:d_{1}\geq\dots\geq d_{n},\ \sum_{i=1}^{n}d_{i}\ \text{is even},\ \sum_{i=1}^{k}d_{i}\leq k(k-1)+\sum_{i=k+1}^{n}\min\{k,d_{i}\}\ \text{for all}\ 1\leq k\leq n \}.$$ One can observe that the diameter of $D(n)$ equals $(n-1)\sqrt[]{n}$. The question is:

How to compute the volume of the convex hull of $D(n)$?

This is worked out in section 3 of Stanley's paper "A Zonotope Associated with Graphical Degree Sequences". The answer is $$\operatorname{Vol}(D(n))=\sum_{X}2^{c(X)}$$ where $X$ ranges through all graphs whose connected components contain a unique cycle, which is of odd length, and $c(X)$ is the number of odd cycles.
• Actually, I compute the volume of the convex hull of all ordered degree sequences of length $n$, that is, I don't assume $d_1\leq d_2\leq \cdots \leq d_n$. The question of Kozerenko is completely different and seems to be much more difficult. Nov 5, 2015 at 14:06