Add Edge Label to certain edge in a bipartite graph

Question

From https://mathematica.stackexchange.com/a/109436/70384 , I modified the code to display the bipartite graph I want and how do I label my edge with the edge weight by manually specifying it? I use EdgeLabels -> Automatic which is not what I want.

---

ClearAll["Global`*"]
colsNames = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0};
attrsNames = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13};

len1 = 13;
len2 = 13;
keys1 = {"A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", 
   "M"};
keys2 = {"T1", "T2", "T3", "T4", "T5", "T6", "T7", "T8", "T9", "T10", 
   "T11", "T12", "T13"};
step = 15;
lis1 = Table[{y1, 0}, {y1, step, step*len1, step}];
lis2 = Table[{y1, 30}, {y1, step, step*len2, step}];

vlabels = 
  Join[MapThread[#1 -> 
      Placed[{Style[#, 16, Bold], Style[#2, 14, Bold]}, {Center, 
        Below}] &, {keys1, colsNames}], 
   MapThread[#1 -> 
      Placed[{Style[#, 16, Bold], Style[#2, 14, Bold]}, {Center, 
        Above}] &, {keys2, attrsNames}]];

Options[bpGraph] = {VertexLabelStyle -> Large, VertexStyle -> Red, 
   EdgeStyle -> Thick, VertexSize -> {"Scaled", .06}, 
   ImageSize -> Medium, VertexCoordinates -> Automatic, 
   VertexLabels -> Placed["Name", Center], EdgeLabels -> Automatic};

bpGraph[nds_, edgs_, opts : OptionsPattern[{bpGraph, Graph}]] := 
 Graph[nds, edgs, 
  Sequence @@ 
   FilterRules[Join[{opts}, Options[bpGraph]], Options[Graph]]]

SetOptions[bpGraph, VertexCoordinates -> Join[lis1, lis2], 
  VertexLabels -> vlabels];

after = bpGraph[
  Join[keys1, keys2], {"A" -> "T12" , "B" -> "T3", "C" -> "T5", 
   "D" -> "T9", "E" -> "T5", "F" -> "T6", "G" -> "T7", "H" -> "T8", 
   "I" -> "T9", "J" -> "T10", "K" -> "T4", "L" -> "T12", 
   "M" -> "T13"}]

This is the modified source code. I would like to specify my edge in this way, for example, EdgeLabels -> {1 <-> 2 -> 45} where 45 is my weight. I don't know how to type the symbol connecting 1 and 2, usually I just copy and paste the middle symbol.

Answer 1

You can typeset a directed edge between nodes x and y as x -> y or x \[DirectedEdge] y or DirectedEdge[x, y], and an undirected edge as x <-> y or x \[UndirectedEdge] y or UndirectedEdge[x, y].

When used with rules we need to parenthesize directed edges as in EdgeLabels -> {(x -> y) ->5} to force correct parsing:

bpGraph[Join[keys1, keys2], {"A" -> "T12", "B" -> "T3", "C" -> "T5", 
   "D" -> "T9", "E" -> "T5", "F" -> "T6", "G" -> "T7", "H" -> "T8", 
   "I" -> "T9", "J" -> "T10", "K" -> "T4", "L" -> "T12", "M" -> "T13"},
  ImageSize -> Large,
  EdgeLabels -> {("B" -> "T3") -> 2, "C" \[DirectedEdge] "T5" -> 3, 
     DirectedEdge["E", "T5"] -> 4}]

For undirected edges we don't need parantheses

bpGraph[Join[keys1, keys2], {"A" -> "T12", "B" <-> "T3", "C" <-> "T5",
   "D" -> "T9", "E" <-> "T5", "F" <-> "T6", "G" -> "T7", "H" -> "T8", 
  "I" -> "T9", "J" -> "T10", "K" -> "T4", "L" -> "T12", "M" -> "T13"},
 ImageSize -> Large,
 EdgeLabels -> {"B" <-> "T3" -> 2, "C" \[UndirectedEdge] "T5" -> 3, 
  UndirectedEdge["E", "T5"] -> 4}]

Note: The reason the form EdgeLabels ->{ 1 -> 2 -> "label"} throws an error is that the expression 1 -> 2 -> "label" is parsed as 1 -> (2 -> "label") as can be seen using the FullForm of that expression:

1 -> 2 -> "label"// FullForm

Rule[1, Rule[2, "label"]]