VertexLabels, EdgeLabels and the direction of the arrows on a directed graph

Question

I want to have three network types. My plan is to put them together in a figure file via marking them with (a) (b) and (c). The figures that I want to have, have some problems. Here is my code

Graph[{2 -> 1, 3 -> 2, 4 -> 3, 5 -> 4}, EdgeStyle -> Arrowheads[.04], VertexLabels -> Placed["Name", Center], VertexSize -> 0.5, ImagePadding -> 20, VertexStyle -> White, EdgeStyle -> Directive[Blue], VertexShapeFunction -> "Square"]

Graph[{1 \[DirectedEdge] 2, 1 \[DirectedEdge] 3, 2 -> 4, 2 -> 5}, EdgeStyle -> Arrowheads[.04], VertexLabels -> Placed["Name", Center], VertexSize -> 0.2, VertexStyle -> White, EdgeStyle -> Blue, VertexShapeFunction -> "Square"]

Graph[{1 \[DirectedEdge] 2, 1 \[DirectedEdge] 3, 1 -> 4, 1 -> 5}, EdgeStyle -> Arrowheads[.04], VertexLabels -> Placed["Name", Center], VertexSize -> 0.35, VertexStyle -> White, EdgeStyle -> Blue, VertexShapeFunction -> "Square"]

The first graph is a example of a tandem graph, the second is a tree and the last one is a parallel. The problems are as follows:

1. I want to put all three figures side by side, therefore the first serial one needs to be vertical not horizontal. The direction of the arrows is correct in the figure.

2. For all graphs, the names of the square boxes need to be $S_1$, $S_2$... intead of only numbers.

3. The length of the edges must be the same. For example second figure has very long edges compared to others

4. The Tree (second) and the parallel (third) networks should have reversed directed edges. The central node should always be $S_1$ and the arrows need to direct that node.

5. The size of the squares need to comparable. I was roughly able to do this.

6. Every node (except node number 1) needs to have a label $y_i$ on the square, where $i$ is the number of the node

7. Every edge needs to be labeled with $u_i$ where $i$ is the number of the outgoing edge. The edge labels should not touch the edges, e.g. can be to the left of the edges.

8. Edges can have all blue colors or black colors. I fail to change the color of the edges. The current color is blue like, not a real blue for example.

9. Node number 1 must be in red color for all graphs.

Thanks in advance for your help.

Answer 1

vlabels = Join[{1 -> Placed[Style[Subscript["S", ToString@1], 18], Center]}, 
                # -> Placed[{Style[Subscript["S", ToString@#], 18], 
                      Style[Subscript["y", ToString@#], 14]}, {Center, Above}] & /@ Range[2, 5]];

elF = Property[#, {EdgeStyle -> Directive[Opacity[1], Arrowheads[Large],Blue], 
    EdgeLabels -> Placed[Style[Subscript["u", ToString@#[[1]]], 14], {1/2, {2, 1}}]}] &;
el1 = elF /@ {2 -> 1, 3 -> 2, 4 -> 3, 5 -> 4};
el2 = elF /@ (Reverse /@ {1 -> 2, 1 -> 3,  2 -> 4, 2 -> 5});
el3 = elF /@ (Reverse /@ {1 -> 2, 1 -> 3,  1 -> 4, 1 -> 5});

style = { VertexLabels -> vlabels, VertexSize -> {"Scaled", .1}, ImagePadding -> 30, 
  VertexStyle -> {1 -> Red, ## & @@ Thread[Range[2, 5] -> White]},
  VertexShapeFunction -> "Square",
  GraphLayout -> {"LayeredEmbedding", "RootVertex" -> 1,
                   LayerSizeFunction -> (2 &), "LeafDistance" -> 2}};

With these settings we address almost all of the eight requirements. Adjusting the image sizes manually we get:

g1 = Graph[el1,  style /. Above -> After, ImageSize -> {200, 400}];
{g2, g3} = Graph[#, style, ImageSize -> #2] & @@@ {{el2, {400, 400}}, {el3, {500, 500}}};
Row[{g1, g2, g3}]