Biology Reference
In-Depth Information
The
k × n
matrix product
E
T
H
aggregates the mating structure for each
genotype. Without loss of generality, let
E
(1
,i
)
=
E
(2
,i
)
=
...
=
E
(
t,i
)
=1
and
E
(
t
+1
,i
)
=
E
(
t
+2
,i
)
=
...
=
E
(
m,i
)
=0. Then the
i
th row of
E
T
H
is
η
(
v
1
)+
η
(
v
2
)+
...
+
η
(
v
t
). From the definition of a diversity graph we know
there are
t/
2 disjoint pairs, (
v
p
,v
q
), with
p
and
q
no greater than
t
, such that
η
(
v
p
)+
η
(
v
q
)=
γ
(
w
i
). This means that the
i
th row of
E
T
H
is (
t/
2)
γ
(
w
i
).We
have just established the following result.
then
E
T
H
Theorem
6.2.
If
(
V
,
W
,
E
,η,γ
)
is
a
diversity
graph,
=
diag
2
E
T
e
G
.
The matrix equation in Theorem 6.2 succinctly separates the structure of the
graph, explained by
E
, from the labeling of the graph, explained by
H
and
G
.
Unfortunately, satisfying the matrix equation does not guarantee the graph is a
diversity graph because the aggregated information ignores the need of a mating
structure. As an example
11
−
1
−
1
−
111
−
1
E
T
H
=(1111)
−
1
−
1
−
11
1
−
111
=(2)(
0000
)
= diag
1
2
E
T
e
G.
This labeling of
K
4
,
1
does not lead to a diversity graph since no pair of haplotypes
(no two rows of
H
) add to form the single genotype (the row of
G
).
We conclude this section with a discussion of a logical operator that helps
address the failure of Theorem 6.2 to characterize graphs with the stated matrix
equation. The
logical join
of a sequence of matrices is determined by the logical
operator “or” over each component of these matrices. The component-wise logical
join is defined so that 0
A
1
,A
2
,...,A
s
}
is a
logical decomposition
of
A
if
A
is the logical join of the matrices in this set,
denoted:
∨
0=0, 0
∨
1=1,and1
∨
1=1.Theset
{
A
i
=
A
1
A
2
A
s
=
A,
∨
∨
...
∨
1
≤i≤s
where we assume that all matrix elements are 0 or 1. For example, the matrices
on the left are a logical decomposition of the matrix on the right,
1100
1111
.
Such decompositions are used in the next section to characterize the graphs that
support diversity.
1010
1100
0101
=
1100
∨
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