# Minimum of sums over degree products in a directed acyclic graph

My problem is the following: we have a graph $$G=(V,E)$$. Having a total ordering $$\eta$$ of the nodes, we give a direction to the edges such that $$(u,v)$$ is directed from $$u$$ to $$v$$ iif $$\eta(u) < \eta(v)$$. Therefore, the directed graph obtained is a directed acyclic graph (DAG).

In this directed graph, is it true that the minimum over all orderings of $$\sum _{i \in V} d^+(i)d^+(i)$$ is greater or equal to the minimum over all orderings of $$\sum _{i \in V} d^+(i)d^-(i)$$? ($$d^+(i)$$ is the out-degree of $$i$$ and $$d^-(i)$$ is its in-degree)

My impression is that it is. There is no obvious minimum ordering for $$\sum _{i \in V} d^+(i)d^-(i)$$, as this problem is known to be NP-hard - see this question on Theoretical Computer Science. I don't know if it is the case for $$\sum _{i \in V} d^+(i)d^+(i)$$. Maybe it is possible to prove that we can always find an ordering which makes $$\sum _{i \in V} d^+(i)d^-(i)$$ smaller or equal to the minimum of $$\sum _{i \in V} d^+(i)d^+(i)$$.