## Combinatorics & Isomorphism

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