1. Computing an Adjacency Matrix D. J. Foreman
Graphs
Computing an Adjacency Matrix
D. J. Foreman

2. The Graph & it's Adj1 A To B A B C D E F 1 D C
From F E

3. Algorithm for Computing Adj2
If Adj[i,1] && Adj[1,j] == 1
 path of length 2 from i to j through node 1
Therefore,
If ANY Adj[i,n] && Adj[n,j] == 1, for all n, then
 a path Length=2 from i to j.
Thus, for [i, j] compute:
Adj[i,1] && Adj[1,j] || Adj[i,2] && Adj[2,j] || … || Adj[i,n] && Adj[n,j]
Repeat for all i and j

4. Computation for 1 pair (A to B)
Let A-F be replaced by 1-6
Find a length 2 path from i=1 to j=2
adj[1,1] && adj[1,2]=0 && 1=>0
adj[1,2] && adj[2,2]=1 && 0=>0
adj[1,3] && adj[3,2]=1 && 1=>1 (13  2)
adj[1,4] && adj[4,2]=1 && 0=>0
adj[1,5] && adj[5,2]=0 && 0=>0
adj[1,6] && adj[6,2]=0 && 1=>0
Now "or" these together  Adj[1,2]
So  a path of length 2 from A to B (through C) See the red edges in previous slide

5. Adj[i,1] && Adj[1,j] || Adj[i,2] && Adj[2,j] || … || Adj[i,n] && Adj[n,j]3 4 5 6 1 2 3 4 5 6
Adj[1]
Adj[2]

6. Key Points Adj (by itself) only tells if a path EXISTS
Does NOT tell what the path IS
Adj1 * Adj2  Adj3 (path of length 3)
Computations describe edges in path
Same as regular matrix multiply, but replace
* with &&
+ with ||