I am reading this paper, and I am a bit stuck on the definitions.
What exactly is $H/\text{Aut}(H)$ and are the elements in that last "set of interest", the $O_{H,i}^V$ just the set of vertices that vertex $i$ can be mapped to by the automorphism? If that is so, then $d_H = \text{Card}(V_H)$ ? I presume the automorphism acts on all vertices mapping them to different other vertices within the graph?
Contrary to what the wording in that quote suggests, the orbits and quotient are being defined for all automorphisms at once, not for just one particular automorphism.
So an orbit of a vertex $v$ is the set of vertices that some automorphism can map $v$ to; they seem to be writing it as $O^V_{H,i}$. Here $i$ is the same for each $v$ in the orbit. Then the graph $H/\text{Aut}(H)$ has the $O^V_{H,i}$ as its vertices; the vertices $O^V_{H,i}$ and $O^V_{H,j}$ are connected by an edge whenever any $H$-vertex in $O^V_{H,i}$ is connected to some $H$-vertex in $O^V_{H,j}$. Since the automorphisms act on the edges too, it is enough to check that one vertex in $O^V_{H,i}$ is connected to a vertex in $O^V_{H,j}$.