When I was working on my PhD dissertation, I came across a physical situation involving nodes and flows between them. It turned out that I was working with a complete oriented graph $K_n$ (all nodes are connected to each other), and I needed to calculate the pseudoinverse of its incidence matrix T, i.e. the rectangular matrix N(V) x N(E) where N(E) = n is the number of edges and N(V) is the number of vertices, with matrix element 1 if the edge enters the vertex, -1 if the edge leaves the vertex, and 0 otherwise.

To my surprise the pseudoinverse turns out to be proportional to the transpose of the incidence matrix! Specifically

$T^{+} (K\_n) = \frac{1}{n} T^{\prime} (K\_n)$

where the prime denotes transposition.

My question is:

A) What other graphs $G$, if any, have this property, i.e. that

$T^+(G) \propto T^\prime(G)$,

or some suitable generalization thereof, and

B) How can I show that this result is invariant of orientation? (I determined empirically is certainly true for all possible orientations of the small complete graphs, and I haven't been able to find a counterexample, but I don't have a proof of this statement yet)

I'm not a professional mathematician, so any thoughts would be welcome.

One class of generalizations that is possible (but not so interesting IMO) are disconnected graphs where each subgraph is a complete graph, i.e.

$G = K_{n_1} \oplus K_{n_2} \oplus \dots \oplus K_{n_m}$

In this case one gets

$T^+(G) = \frac{1}{n_1} T^\prime(K_{n_1}) \oplus \frac{1}{n_2} T^\prime(K_{n_2}) \oplus \dots \oplus \frac{1}{n_m} T^\prime(K_{n_m})$

which is not really that exciting, but perhaps could point the way to a more interesting generalization.

P.S. There is a short proof of the first fact, which relies on the fact that the complete graph has a Laplacian of the form

$\Delta = n \mathbf{I} - \mathbf{1} \mathbf{1}^\prime$

where $\mathbf{I}$ is the n x n identity matrix and $\mathbf{1}$ is a column vector of ones.

With this fact, together with knowing that the column sums of $T$ are all zero, it is straightforward to show that $T^\prime(K_n)/n$ satifies all the Moore-Penrose conditions for the pseudoinverse.

P.P.S. If anyone is interested in the physical context, here is where it came from.