Automorphism proof (graph theory)

[TheMathNoob]

Homework Statement

The problem is attached and it's part A. There is no need to put problem 4 hence the problem is fully explained in the file attached

Homework Equations

Zk is mod k basically.

The Attempt at a Solution

I know that we have to prove that the transformation is onto,one to one and preserves adjacency.
It's one to one because
T(s1)=T(s2)
s1+v=s2+v
s1=s2

It's onto because
y=s+v
y-v=s
T(s)=T(y-v)=y-v+v=y
I am not quite sure how to show that it preserves adjacency because I can't apply the concept of hamming distance anymore.

 

[andrewkirk]

Two vertices are adjacent iff they have identical coordinates in all dimensions except for one.

Say $U$ and $V$ are adjacent because their coordinates are identical except for the $j$th one.

For which, if any, of $u\in\{1,2,...,n\}$ are the $u$th coordinates of $T(U)$ and $T(V)$ equal?

 

[TheMathNoob]

Two vertices are adjacent iff they have identical coordinates in all dimensions except for one.

Say $U$ and $V$ are adjacent because their coordinates are identical except for the $j$th one.

For which, if any, of $u\in\{1,2,...,n\}$ are the $u$th coordinates of $T(U)$ and $T(V)$ equal?

Yes except for the jth coordinate?

 

[andrewkirk]

Yes, and what does that tell us about whether T(U) and T(V) are adjacent?

 

[TheMathNoob]

Yes, and what does that tell us about whether T(U) and T(V) are adjacent?

T(U) and T(V) are still adjacent because all their coordinates are the same except for the jth one.

 

[TheMathNoob]

Yes, and what does that tell us about whether T(U) and T(V) are adjacent?

I had another problem. I would be glad if you take a look.