Hi,
I am not a mathematician so bear with me.
I would like to just clarify my understanding of isomorphism using the following 4 adjacency matrices for the following parameters:
Where:
v=10
k=6
t=3
And there are 10 blocks in each of the 4 matrices and all 4 matrices satisfy the parameters.
Matrix A:
{ 1, 2, 3, 4, 6, 7}
{ 1, 2, 3, 5, 8,10}
{ 1, 2, 3, 7, 9,10}
{ 1, 2, 4, 6, 8,10}
{ 1, 3, 4, 5, 6, 9}
{ 1, 4, 5, 7, 8, 9}
{ 2, 4, 5, 6, 9,10}
{ 2, 5, 6, 7, 8, 9}
{ 3, 4, 5, 7, 8,10}
{ 3, 6, 7, 8, 9,10}
Matrix B:
{ 1, 2, 3, 4, 6, 7}
{ 1, 2, 3, 5, 8,10}
{ 1, 2, 3, 7, 9,10}
{ 1, 2, 4, 6, 8,10}
{ 2, 3, 4, 5, 6, 9}
{ 1, 4, 5, 7, 8, 9}
{ 1, 4, 5, 6, 9,10}
{ 2, 5, 6, 7, 8, 9}
{ 3, 4, 5, 7, 8,10}
{ 3, 6, 7, 8, 9,10}
Matrix C:
{ 1, 2, 3, 4, 6, 7}
{ 1, 2, 3, 5, 7,10}
{ 1, 2, 3, 8, 9,10}
{ 1, 2, 4, 6, 8,10}
{ 1, 3, 4, 5, 6, 9}
{ 1, 4, 5, 7, 8, 9}
{ 2, 4, 5, 6, 9,10}
{ 2, 5, 6, 7, 8, 9}
{ 3, 4, 5, 7, 8,10}
{ 3, 6, 7, 8, 9,10}
Matrix D:
{ 1, 2, 3, 4, 6, 7}
{ 1, 2, 3, 5, 7,10}
{ 1, 2, 3, 8, 9,10}
{ 1, 2, 4, 6, 8,10}
{ 2, 3, 4, 5, 6, 9}
{ 1, 4, 5, 7, 8, 9}
{ 1, 4, 5, 6, 9,10}
{ 2, 5, 6, 7, 8, 9}
{ 3, 4, 5, 7, 8,10}
{ 3, 6, 7, 8, 9,10}
Based on my understanding of isomorphism NONE of these 4 matrices (A,B,C,D) are isomorphic because of the following comparing 2 matrices at a time:
For Matrix A:
Matrix A and B are not isomorphic because the 1 and 2 are relabeled only in blocks 5 & 7 (should be blocks 5,6,7 and 8)
Matrix A and C are not isomorphic because the 7 and 8 are relabeled only in blocks 2 & 3 (should be blocks 1,2,3 and 4)
Matrix A and D are not isomorphic because the 1 and 2 are relabeled only in blocks 5 & 7 (should be blocks 5,6,7 and 8) and also because 7 & 8 are relabeled only in blocks 2 & 3 (should be blocks 1,2,3 and 4).
For Matrix B:
Matrix B and C are not isomorphic because the 7 and 8 are relabeled only in blocks 2 & 3 (should be blocks 1,2,3 AND 4) and also because 1 & 2 are relabeled only in blocks 5 & 7 (should be blocks 5,6,7 and 8).
Matrix B and D are not isomorphic because the 7 and 8 are relabeled only in blocks 2 & 3 (should be blocks 1,2,3 and 4)
For Matrix C:
Matrix C and D are not isomorphic because the 1 and 2 are relabeled only in blocks 5 & 7 (should be blocks 5,6,7 and 8)
Is this right?
Are the 4 matrices (A,B,C,D) all non-isomorphic?
And if any are isomorphic, can you explain why they are?
Furthermore, would most software that compares 2 graphs to see if they are isomorphic catch these subtelties?
Thanks