How to merge non disjoint graphs through their adjacency matrices?

Question

I'm a second year physics undergraduate student and a novice to programming and to Mathematica. I've started self studying graph theory some months ago and I'm getting acquainted with it and how to use Mathematica to work with graphs.

One of the problems I'm trying solve is how to code the operations for merging non disjoint graphs, using their adjacency matrices, for example, if I have these adjacency matrices:

I realized that one way to create the joint adjacency matrix was to first dispose de matrices on the diagonal of a 9x9 matrix, like (1) and then transferring the values of the repeated columns to the first columns where that vertex appeared and finally deleting the duplicated columns, thus generating a matrix like (2), which in turn allows me to use functions like Graph and AdjacencyGraph to create the intended graph.

These operations are relatively simple to perform on a spreadsheet software like Excel, but I found out that I couldn't crack how to do it on Mathematica, even though I know how to do the basic linear algebra and matrix algebra in the Wolfram Language.

I know that this is probably not the kind of answer this community is intended to have, but I don't know anywhere else to ask for help.

Thanks in advance,

Answer 1

Perhaps this could be useful.

You can create your adjacency matrices as lists of lists.

(*Adjacency Matrices*)
M1 = {{0, 1, 1}, {1, 0, 0}, {1, 1, 0}};
M2 = {{0, 1, 1}, {1, 0, 0}, {1, 1, 0}};
M3 = {{0, 0, 1}, {1, 0, 1}, {1, 1, 0}};

Then your labels of the "v's"

(*Labels*)
vlabel1 = {1, 2, 3};
vlabel2 = {4, 1, 5};
vlabel3 = {6, 4, 7};

Then using sparse array you can set particular positions in your matrix to be one according to your adjacency matrix, and use MatrixForm to visualise. (The final matrix however is different to yours). Hope this helps

(*Rules for Sparse Array*)

r1 = Map[# -> 1 &, Map[vlabel1[[#]] &, Position[M1, 1]]];
r2 = Map[# -> 1 &, Map[vlabel2[[#]] &, Position[M2, 1]]];
r3 = Map[# -> 1 &, Map[vlabel3[[#]] &, Position[M3, 1]]];
SparseArray[Join[r1, r2, r3]] // MatrixForm

Answer 2

You can also use the GraphUnion of the AdjacencyGraphs obtained using the three adjacency matrices and their vertex lists:

am1 = {{0, 1, 1}, {1, 0, 0}, {1, 1, 0}};
am2 = {{0, 1, 1}, {1, 0, 0}, {1, 1, 0}};
am3 = {{0, 0, 1}, {1, 0, 1}, {1, 1, 0}};
vl1 = {1, 2, 3};
vl2 = {4, 1, 5};
vl3 = {6, 4, 7};

Get three AdjacencyGraphs using the vertex lists and adjacency matrices defined above:

{g1, g2, g3} = (AdjacencyGraph @@@ {{vl1, am1}, {vl2, am2}, {vl3, am3}});

Row[Labeled[SetProperty[#, {VertexShapeFunction->"Name", ImageSize->200}],  #2, Top] & @@@
  {{g1, "g1"}, {g2, "g2"}, {g3, "g3"}}]

Take the GraphUnion of g1, g2, g31:

gu = GraphUnion[g1, g2, g3];

SetProperty[gu, {VertexShapeFunction -> "Name", ImageSize -> 500}]

Get the AdjacencyMatrix of gu:

am = AdjacencyMatrix[gu]

SparseArray["<"15">",{7,7}]

am // MatrixForm