310
Solutions to Exercises
(b) See Exercise 18(b) of Section 11.2. The score sequences, left to right, are 3,2,1,0, 3,1,1,1,
2,2,2,0 and 2, 2, 1,
1.
Only the leftmost tournament is transitive.
2. [BB] Since for each pair of (distinct) vertices
Vi, Vj
precisely one of
ViVj, VjVi
is an arc, in the adja
cency matrix
A,
for each
i
=I
j,
precisely one of
aij, aji
is
Thus,
A
+
AT
has o's on the diagonal
and l's in every offdiagonal position.
3. (a) If
A
is the adjacency matrix of a tournament, then
A
+
has o's on the diagonal and l's
everywhere else, by Exercise 2. This matrix is
J

I.
On the other hand, if
A
+
=
J

I
for some adjacency matrix
then
2aii
=
0, so each diagonal entry of
A
is O. Also, if
i
aij
+
aji
=
1 with each
either 0 or 1 says that exactly one of these entries is 1, while the other
is O. Thus
A
is the adjacency matrix of a tournament.
(b) Let
A
be the adjacency matrix of a tournament and let
B
=
AT.
Then
B
+
BT
=
+
so
B
is also the adjacency matrix of a tournament, by Exercise 3(a)
(c) Using Exercise 3(a), it suffices to show that
P ApT
=
J

We know
A
=
J

I,
so
=
p
J
pT
_
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 Summer '10
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 Graph Theory, Matrices, Trigraph, Arc, adjacency matrix, Adjacency list

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