Name of an operation on graphs Let $G$ and $H$ be two possibly directed, non necessarily simple, vertex-labelled graphs with respective adjacency matrices $A_G$ and $A_H$ and $V(G)=V(H)$.
1) What is the name of the graph $M$ with adjacency matrix $A_M=A_HA_G$?
2) In the unlikely event that I am the first to think about using that operation, which symbols should I NOT use to denote it in order to avoid confusion with other graph products?
UPDATE: I have now asked this question on MathOverflow as well.
 A: I believe this is known as Graph Squaring. See also here.
A: As described in Brualdi and Cvetkovic's Combinatorial Approach to Matrix Theory,  rather than associating a directed graph in the usual way to a matrix, it's more natural to associate the König digraph for the purposes of thinking about matrix multiplication.  For an $m \times n$ non-negative integer matrix $A$, the König digraph is a bipartite digraph with vertex classes $X$ (with $n$ elements) and $Y$ (with $m$ elements) such that the number of edges from $x \in X$ to $y \in Y$ is $A_{yx}$.  Composition of matrices then corresponds to "plugging" one digraph into another, and whatever purpose you want to multiply adjacency matrices for, I think it will be more natural (or at least equivalent) phrased in this language instead.
What I've described is a category whose objects are the non-negative integers $[n], n \ge 0$ and where a morphism from $[n]$ to $[m]$ is a collection of arrows from an $n$-element set to an $m$-element.  This category is a certain refinement of the category of finite sets and relations and is a very natural setting for certain combinatorial arguments (as well as for, as the book says, a combinatorial approach to matrix theory).  I don't know if it has a name, though.
A: Maybe it would be helpful... from position that $M$ is a hypergraph, value of matrix element $A_M(i,j)$ seems to be equal to the number of edges from vertex i to vertex j with length 2 (minus edges from j to i). But I'm not sure, if $A_M$ is adjacency matrix for hypergraph $M$... Especially after googling "hypergraph adjacency matrices"
