Vertices connection

From your question, I am not sure how much information you have. In any case, you can create a map of vertices to the edges they connect with:

TopTools_IndexedDataMapOfShapeListOfShape mapVertexEdge;
TopExp::MapShapesAndAncestors(myShape, TopAbs_VERTEX, TopAbs_EDGE, mapVertexEdge);

I don't believe there is a CASCADE function to get a list of the
connectivity between edges. I use the following:

TopTools_IndexedDataMapOfShapeListOfShape mapEdgeEdge;
// create an empty list of edges
TopTools_ListOfShape listEdge;

// visit the edges in the geometry shape
TopExp_Explorer exEdge;
for (exEdge.Init(myShape, TopAbs_EDGE); exEdge.More(); exEdge.Next()) {
const TopoDS_Shape &edge = exEdge.Current();

// visit the vertexes on this edge
TopExp_Explorer exVertex;
for (exVertex.Init(edge, TopAbs_VERTEX);
exVertex.More(); exVertex.Next()) {
const TopoDS_Shape &vertex = exVertex.Current();

// get the list of edges attached to this vertex
const TopTools_ListOfShape &listAtt = mapVertexEdge.FindFromKey(vertex);

// loop through the edge list
TopTools_ListIteratorOfListOfShape itEdge;
for (itEdge.Initialize(listAtt); itEdge.More(); itEdge.Next()) {
// if (not the current edge) add it to the list
if (itEdge.Value() != edge) {
// first, make sure it is not already in the list
TopTools_ListIteratorOfListOfShape itCheck;
for (itCheck.Initialize(listEdge); itCheck.More(); itCheck.Next()) {
if (itCheck.Value() == itEdge.Value()) break;
}
if (!itCheck.More()) listEdge.Append(itEdge.Value());
}
}
}
// store the list for this face
mapEdgeEdge.Add(edge, listEdge);

// clear the list for the next iteration
listEdge.Clear();
}