> < ^ Date: Tue, 11 May 1999 13:46:45 +0100 (BST)
> < ^ From: Leonard Soicher <l.h.soicher@qmul.ac.uk >
^ Subject: On (collapsed) adjacency matrices in GRAPE

Dear GAP-forum,

Drew Krause wrote:

Apologies for the earlier post; 'CollapsedAdjacencyMat(g)' will give an

This is not correct in general.

In GRAPE (2.3 and above), if gamma.group is transitive on the
vertices of gamma then `CollapsedAdjacencyMat( gamma )' returns the
collapsed adjacency matrix of gamma, collapsed with respect to
Stabilizer(gamma.group,1), which may or may not be trivial.

However, `CollapsedAdjacenyMat( Group(()), gamma )' will always return
the (uncollapsed, ordinary) adjaceny matrix of the graph gamma in
GRAPE. (See the documentation for CollapsedAdjacencyMat to see what
this function does in general.)

For example (using GRAPE 4.0 under GAP4b5, but GRAPE 2.31 behaves
similarly):

```gap> J:=JohnsonGraph(4,2);
rec( isGraph := true, order := 6,
group := Group([ (1,4,6,3)(2,5), (2,4)(3,5) ]),
schreierVector := [ -1, 2, 1, 1, 1, 1 ], adjacencies := [ [ 2, 3, 4, 5 ] ],
representatives := [ 1 ],
names := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ], [ 3, 4 ] ],
isSimple := true )
[ [ 0, 4, 0 ], [ 1, 2, 1 ], [ 0, 4, 0 ] ]
[ [ 0, 1, 1, 1, 1, 0 ], [ 1, 0, 1, 1, 0, 1 ], [ 1, 1, 0, 0, 1, 1 ],
[ 1, 1, 0, 0, 1, 1 ], [ 1, 0, 1, 1, 0, 1 ], [ 0, 1, 1, 1, 1, 0 ] ]
```

Drew Krause continues:

Another question: has anyone developed a 'grape' algorithm for
discovering a graph's Eulerian path, if it exists?

Not that I know of.

Regards, Leonard Soicher.

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