Here's a simple function that can convert Graphics to a graph. With this, you can draw a simple graph with the Drawing Tools palette (http://reference.wolfram.com/language/tutorial/InteractiveGraphicsPalette.html) and then copy the graphics into this function:
graphicsToGraph[gr : _Graphics | _Graphics3D] := Module[{
pts = Join @@ Cases[gr,
Point[arg_] :> Replace[Setting[arg], lst : {__?NumericQ} :> {lst}],
DirectedInfinity[1]
],
edges1 = Cases[gr,
Line[lst_] :> UndirectedEdge @@ Setting[lst][[{1, -1}]],
DirectedInfinity[1]
],
edges2 = Cases[gr,
Arrow[lst_] :> DirectedEdge @@ Setting[lst][[{1, -1}]],
DirectedInfinity[1]
],
nf, vertices
},
vertices = Range[Length[pts]];
Condition[
nf = Nearest[pts -> "Index"];
Graph[
vertices,
Map[First @ nf[#, 1]&, Join[edges1, edges2], {2}],
VertexCoordinates -> Thread[vertices -> pts]
]
,
Length[pts] > 0
]
];
The function will look for Point, Line and Arrow primitives and then join them up in the most sensible way it can to make a graph. Only the begin and end points of lines/arrows will be kept, so you can draw segmented lines to avoid spaghetti.
Getting the adjacency matrix, vertex list and edge lists is easy from that point:
AdjacencyMatrix[graph]
VertexList[graph]
EdgeList[graph]