D. C. Mayer
DOI:
10.4236/apm.2018.81006 87 Advances in Pure Mathematics
with
( )
1
1
i
j Cv≤≤
denote
the
capable
children
of
i
v
,
for
each
(
)
1
2
e
i Cm
≤≤
.
1) If the tree is of depth
( )
dp 1=
, then
( )
( )
1
#
e
e Nm
=
(5.9)
2) If the tree is of depth
(
)
dp 2
=
, then
(
)
( )
(
)
(
)
1
11
2
#
e
Cm
ei
i
e Nm Nv
=
= +
∑
(5.10)
3) If the tree is of depth
(
)
dp 3
=
, then
( )
( )
( )
( )
( )
( )
11
1 1 1,
21
#
ei
Cm Cv
e i ij
ij
e Nm Nv Nv
= =
=++
∑∑
(5.11)
Proof
. Put
(
)
: e
=
. Generally, we have
Lyr Lyr
e ed
+
=
∪∪
with
(
)
: dpd
=
, according to Lemma 5.1.
If
( )
dp 1=
, then
1d ≤
and
1
Lyr Lyr
ee+
=
∪
. We have
{ }
Lyr
ee
m=
and
( )
11
#Lyr 1
ee
Nm
+
= −
, since the next mainline vertex
1e
m
+
is one of the
( )
1 e
Nm
children of
e
m
but does not belong to
. Thus, we obtain
( ) ( )
11 1
# #Lyr #Lyr 1 1
ee e e
Nm Nm
+
= + =+ −=
.
If
( )
dp 2=
, then
2d ≤
and
12
Lyr Lyr Lyr
ee e++
=
∪∪
, where
#Lyr 1
e
=
and
( )
11
#Lyr 1
ee
Nm
+
= −
as before, and
( )
( )
1
21
=2
#Lyr
e
Cm
ei
i
Nv
+
=
∑
.
Therefore,
( )
( )
( )
1
21 1
2
# = #Lyr #Lyr
e
Cm
e ee i
i
Nm Nv
+
=
++ = +
∑
.
If
( )
dp 3=
, then
3d ≤
and
3
Lyr Lyr
ee+
=
∪∪
, where
#Lyr 1
e
=
,
( )
11
#Lyr 1
ee
Nm
+
= −
,
(
)
(
)
1
21
2
#Lyr
e
Cm
ei
i
Nv
+
=
=
∑
as before, and
( ) ( )
( )
11
3 1,
21
#Lyr
ei
Cm Cv
e ij
ij
Nv
+
= =
=
∑∑
. Thus,
( )
(
)
(
)
(
)
(
)
( )
11
3 1 1 1,
21
# #Lyr #Lyr
ei
Cm Cv
e e e i ij
ij
Nm Nv Nv
+
= =
= ++ = + +
∑∑
Remark 5.1.
In
Theorem
5.1,
item
(1)
is
included
in
item
(2),
since
( )
dp 1=
implies
( )
1
1
e
Cm
=
,
and
item
(2)
is
included
in
item
(3),
since
( )
dp 2=
implies
( )
1
0
i
Cv=
,
for
all
( )
1
2
e
i Cm≤≤
.
Corollary 5.1.
Under
the
same
assumptions
as
in
Theorem
5.1,
the
width
of
the
coclass
tree
,
in
dependence
on
the
depth
( )
: dpd =
and
the
periodicity
(
)
,
∗
,
is
generally
given
by
( )
{ }
wd max #Lyr |
n
n nn d
∗ ∗∗
= << + ++
(5.12)
For assigned small values of the depth
3d ≤
, the width can be expressed in
terms of descendant numbers in the following manner:
1) If the tree is of depth
1d =
, then
( )
( )
{ }
11
wd max | 1
n
Nm n n n
− ∗ ∗∗
= +≤ ≤ + +
(5.13)
2) If the tree is of depth
2d =
, then
( )
wd
is the maximum among the
number
( )
1 n
Nm
∗
and all expressions
( )
( )
( )
( )
12
11 1 2
2
n
Cm
n in
i
Nm Nvm
−
−−
=
+
∑
(5.14)
where
n
runs from
2n
∗
+
to
1n
∗∗
+ ++
.
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