In this section, we determine the maximum value of Estrada index among the set of 3uniform linear hypertrees.
Proof.
Firstly, we construct a mapping
φ
from
W
k
(
S
m
3
;
v
2
)
to
W
k
(
S
m
3
;
v
1
)
. For
W
∈
W
k
(
S
m
3
;
v
2
)
, let
φ
(
W
)
be the closed walk obtained from
W
by replacing
v
1
by
v
2
and
v
2
by
v
1
. Obviously,
φ
(
W
)
∈
W
k
(
S
m
3
;
v
1
)
and
φ
is a bijection.
Secondly, we construct a mapping
ξ
from
W
k
(
S
m
3
;
v
2
)
to
W
k
(
S
m
3
;
v
1
)
. For
W
∈
W
k
(
S
m
3
;
v
2
)
, we consider the following cases.
Case 1. Suppose
W
does not pass the edge
v
1
v
t
for
t
≥
4
; then
ξ
(
W
)
=
φ
(
W
)
.
Case 2. Suppose
W
passes the edge
v
1
v
t
for
t
≥
4
. For
W
∈
W
k
(
S
m
3
;
v
2
)
, we may uniquely decompose
W
into three sections
W
1
W
2
W
3
, where
W
1
is the longest
(
v
2
,
v
1
)
section of
W
without
v
t
,
W
2
is the internal longest
(
v
t
,
v
t
′
)
section of
W
for
t
′
≥
4
, and the last
W
3
is the remaining
(
v
1
,
v
2
)
section of
W
not containing
v
t
. We consider the following three subcases.
Case 2.1. If both
W
1
and
W
3
contain the vertex
v
3
, we may uniquely decompose
W
1
into two sections
W
11
W
12
and decompose
W
3
into two sections
W
31
W
32
, where
W
11
is the shortest
(
v
2
,
v
3
)
section of
W
1
,
W
12
is the remaining
(
v
3
,
v
1
)
section of
W
1
,
W
31
is the longest
(
v
1
,
v
3
)
section of
W
3
, and
W
32
is the remaining
(
v
3
,
v
2
)
section of
W
3
.
Let
ξ
(
W
)
=
ξ
(
W
11
)
ξ
(
W
12
)
ξ
(
W
2
)
ξ
(
W
31
)
ξ
(
W
32
)
, where
ξ
(
W
12
)
=
W
12
,
ξ
(
W
2
)
=
W
2
,
ξ
(
W
31
)
=
W
31
,
ξ
(
W
11
)
is a
(
v
1
,
v
3
)
walk obtained from
W
11
replacing
v
1
by
v
2
and
v
2
by
v
1
, and
ξ
(
W
32
)
is a
(
v
3
,
v
1
)
walk obtained from
W
32
replacing
v
1
by
v
2
and
v
2
by
v
1
.
Case 2.2. If
W
1
contains the vertex
v
3
and
W
3
does not contain
v
3
, let
ξ
(
W
)
=
ξ
(
W
1
)
ξ
(
W
2
)
ξ
(
W
3
)
, where
ξ
(
W
2
)
=
W
2
,
ξ
(
W
1
)
is a
(
v
1
,
v
1
)
walk obtained from
W
1
replacing its first vertex
v
2
by
v
1
v
2
, and
ξ
(
W
3
)
is a
(
v
1
,
v
1
)
walk obtained from
W
3
replacing its last two vertices
v
1
v
2
by
v
1
.
Case 2.3. If
W
1
does not contain the vertex
v
3
, let
ξ
(
W
)
=
ξ
(
W
1
)
ξ
(
W
2
)
ξ
(
W
3
)
, where
ξ
(
W
2
)
=
W
2
,
ξ
(
W
1
)
is a
(
v
1
,
v
1
)
walk obtained from
W
1
replacing its first two vertices
v
2
v
1
by
v
1
, and
ξ
(
W
3
)
is a
(
v
1
,
v
1
)
walk obtained from
W
3
replacing its last vertex
v
2
by
v
2
v
1
.
For example, in star
S
3
3
on
7
vertices and
3
triangles,
W
=
v
2
v
3
v
1
v
2
v
1
v
3
v
2
is a closed walk of length 6 of
v
2
not passing the edge
v
1
v
t
. By Case 1, we have
(5)
ξ
(
W
)
=
v
1
v
3
v
2
v
1
v
2
v
3
v
1
.
W
′
=
v
2
v
3
v
1
v
4
v
5
v
1
v
6
v
7
v
1
v
2
is a closed walk of length 9 of
v
2
passing the edge
v
1
v
t
. By Case 2.2, we get
(6)
ξ
(
W
′
)
=
v
1
v
2
v
3
v
1
v
4
v
5
v
1
v
6
v
7
v
1
.
W
′′
=
v
2
v
1
v
2
v
1
v
4
v
5
v
1
v
2
v
3
v
1
v
6
v
7
v
1
v
3
v
2
is a closed walk of length 14 of
v
2
passing the edge
v
1
v
t
. By Case 2.3, we obtain
(7)
ξ
(
W
′′
)
=
v
1
v
2
v
1
v
4
v
5
v
1
v
2
v
3
v
1
v
6
v
7
v
1
v
3
v
2
v
1
.
Obviously,
ξ
(
W
)
∈
W
k
(
S
m
3
;
v
1
)
,
ξ
is an injective and not a surjective for
n
≥
5
, and
k
≥
1
.
Proof.
Let
W
k
(
G
i
)
(
W
k
(
G
)
,
W
k
(
S
m
3
∪
Q
)
, resp.) be the set of closed walks of length
k
of
G
i
(
G
,
S
m
3
∪
Q
, resp.) for
i
=
1,2
. Then
W
k
(
G
i
)
=
W
k
(
G
)
∪
W
k
(
S
m
3
∪
Q
)
∪
X
i
is a partition, where
X
i
is the set of closed walks of length
k
of
G
i
; each of them contains both at least one edge in
E
(
G
)
and at least one edge in
E
(
S
m
3
∪
Q
)
. So
M
k
(
G
i
)
=

W
k
(
G
)

+

W
k
(
S
m
3
∪
Q
)

+

X
i

=
M
k
(
G
)
+
M
k
(
S
m
3
∪
Q
)
+

X
i

. Thus we need to show the inequality

X
1

<

X
2

.
We construct a mapping
η
from
X
1
to
X
2
and consider the following four cases.
Case 1. Suppose
W
is a closed walk starting from
u
∈
V
(
G
)
in
X
1
. For
W
∈
X
1
, let
η
(
W
)
=
(
W

W
∩
(
S
m
3
∪
Q
)
)
∪
ξ
(
W
∩
(
S
m
3
∪
Q
)
)
; that is,
η
(
W
)
is the closed walk in
X
2
obtained from
W
by replacing its every section in
S
m
3
∪
Q
with its image under the map
ξ
.
Case 2. Suppose
W
is a closed walk starting at
v
1
in
X
1
. For
W
∈
X
1
, we may uniquely decompose
W
into three sections
W
1
W
2
W
3
, where
W
1
is the longest
(
v
1
,
v
2
)
section of
W
without vertices
u
0
,
…
,
u
t
′′
∈
V
(
G
)
,
W
2
is the internal longest
(
u
0
,
u
t
′′
)
section of
W
(for which the internal vertices are some possible vertices in
V
(
G
1
)
), and
W
3
is the remaining
(
v
2
,
v
1
)
section of
W
. Let
η
(
W
)
=
η
(
W
1
)
η
(
W
2
)
η
(
W
3
)
, where
η
(
W
1
)
=
W
1

1
,
η
(
W
3
)
=
W
3

1
, and
η
(
W
2
)
=
(
W
2

W
2
∩
(
S
m
3
∪
Q
)
)
∪
ξ
(
W
2
∩
(
S
m
3
∪
Q
)
)
; that is,
η
(
W
2
)
is a
(
u
0
,
u
t
′′
)
walk from
W
2
by replacing its every section in
S
m
3
∪
Q
with its image under the map
ξ
.
Case 3. Suppose
W
is a closed walk starting from
v
3
or
w
∈
V
(
Q
)
in
X
1
. For
W
∈
X
1
, we may uniquely decompose
W
into three sections
W
1
W
2
W
3
, where
W
1
is the longest
(
v
3
,
v
2
)
(or
(
w
,
v
2
)
)section of
W
without vertices
u
0
,
…
,
u
t
′
′
,
W
2
is the internal longest
(
u
0
,
u
t
′
′
)
section of
W
(for which the internal vertices are some possible vertices in
V
(
G
1
)
), and
W
3
is the remaining
(
v
2
,
v
3
)
(or
(
v
2
,
w
)
)section of
W
without vertices
u
0
,
…
,
u
t
′
′
. We have three subcases.
Case 3.1. If both
W
1
and
W
3
do not pass edge
v
1
v
t
, let
η
(
W
)
=
η
(
W
1
)
η
(
W
2
)
η
(
W
3
)
, where
η
(
W
2
)
=
(
W
2

W
2
∩
(
S
m
3
∪
Q
)
)
∪
ξ
(
W
2
∩
(
S
m
3
∪
Q
)
)
,
η
(
W
1
)
is a
(
v
3
,
v
1
)
(or
(
w
,
v
1
)
)walk obtained from
W
1
replacing
v
1
by
v
2
and
v
2
by
v
1
, and
η
(
W
3
)
is a
(
v
1
,
v
3
)
(or
(
v
1
,
w
)
)walk obtained from
W
3
replacing
v
1
by
v
2
and
v
2
by
v
1
.
Case 3.2. If both
W
1
and
W
3
pass edge
v
1
v
t
, we may anew decompose
W
into five sections
W
1
W
2
W
3
W
4
W
5
, where
W
1
is the longest
(
v
3
,
v
t
)
(or
(
w
,
v
t
)
)section of
W
(which do not contain vertices
u
0
,
…
,
u
t
′
′
),
W
2
is the second
(
v
1
,
v
2
)
section of
W
(for which the internal vertices, if exist, are only possible
v
1
,
v
2
,
v
3
,
w
∈
V
(
Q
)
), the third
W
3
is the internal longest
(
u
0
,
u
t
′
′
)
section of
W
(for which the internal vertices are some possible vertices in
V
(
G
1
)
), the fourth
W
4
is the longest
(
v
2
,
v
1
)
section of
W
(for which the internal vertices, if exist, are only possible
v
1
,
v
2
,
v
3
,
w
∈
V
(
Q
)
), and the last
W
5
is the remaining
(
v
t
′
,
v
3
)
(or
(
v
t
,
w
)
)section of
W
. We have three subsubcases.
Case 3.2.1. If both
W
2
and
W
4
contain the vertex
v
3
, we may uniquely decompose
W
2
into two sections
W
21
W
22
and
W
4
into two sections
W
41
W
42
, where
W
21
is the longest
(
v
1
,
v
3
)
section of
W
2
,
W
22
is the remaining shortest
(
v
3
,
v
2
)
of
W
2
,
W
41
is the shortest
(
v
2
,
v
3
)
section of
W
4
, and
W
42
is the remaining longest
(
v
3
,
v
1
)
section of
W
4
.
Let
η
(
W
)
=
η
(
W
1
)
η
(
W
21
)
η
(
W
22
)
η
(
W
3
)
η
(
W
41
)
η
(
W
42
)
η
(
W
5
)
, where
η
(
W
1
)
=
W
1
,
η
(
W
21
)
=
W
21
,
η
(
W
3
)
=
(
W
3

W
3
∩
(
S
m
3
∪
Q
)
)
∪
ξ
(
W
3
∩
(
S
m
3
∪
Q
)
)
,
η
(
W
42
)
=
W
42
η
(
W
5
)
=
W
5
,
η
(
W
22
)
is a
(
v
3
,
v
1
)
walk obtained from
W
22
replacing
v
1
by
v
2
and
v
2
by
v
1
, and
η
(
W
41
)
is a
(
v
1
,
v
3
)
walk obtained from
W
41
replacing
v
1
by
v
2
and
v
2
by
v
1
.
Case 3.2.2. If
W
2
does not contain the vertex
v
3
, let
η
(
W
)
=
η
(
W
1
)
η
(
W
2
)
η
(
W
3
)
η
(
W
4
)
η
(
W
5
)
, where
η
(
W
1
)
=
W
1
,
η
(
W
3
)
=
(
W
3

W
3
∩
(
S
m
3
∪
Q
)
)
∪
ξ
(
W
3
∩
(
S
m
3
∪
Q
)
)
,
η
(
W
5
)
=
W
5
,
η
(
W
2
)
is a
(
v
1
,
v
1
)
walk obtained from
W
2
replacing its last two vertices
v
1
v
2
by
v
1
, and
η
(
W
4
)
is a
(
v
1
,
v
1
)
walk obtained from
W
4
replacing its first vertex
v
2
by
v
1
v
2
.
Case 3.2.3. If
W
2
contains the vertex
v
3
and
W
4
does not contain vertex
v
3
, let
η
(
W
)
=
η
(
W
1
)
η
(
W
2
)
η
(
W
3
)
η
(
W
4
)
η
(
W
5
)
, where
η
(
W
1
)
=
W
1
,
η
(
W
3
)
=
(
W
3

W
3
∩
(
S
m
3
∪
Q
)
)
∪
ξ
(
W
3
∩
(
S
m
3
∪
Q
)
)
,
η
(
W
5
)
=
W
5
,
η
(
W
2
)
is a
(
v
1
,
v
1
)
walk obtained from
W
2
replacing its last vertex
v
2
by
v
2
v
1
, and
η
(
W
4
)
is a
(
v
1
,
v
1
)
walk obtained from
W
4
replacing its first two vertices
v
2
v
1
by
v
1
.
Case 3.3. If
W
1
passes edge
v
1
v
t
and
W
3
does not pass edge
v
1
v
t
, we may anew decompose
W
into four sections
W
1
W
2
W
3
W
4
, where
W
1
is the longest
(
v
3
,
v
t
)
(or
(
w
,
v
t
)
)section of
W
(which do not contain vertices
u
0
,
…
,
u
t
′
′
),
W
2
is the second
(
v
1
,
v
2
)
section of
W
(for which the internal vertices, if exist, are only possible
v
1
,
v
2
,
v
3
,
w
∈
V
(
Q
)
), the third
W
3
is the internal longest
(
u
0
,
u
t
′
′
)
section of
W
(for which the internal vertices are some possible vertices in
V
(
G
1
)
), and the last
W
4
is the longest
(
v
2
,
v
3
)
(or
(
v
2
,
w
)
)section of
W
(for which the internal vertices, if exist, are only possible
v
1
,
v
2
,
v
3
,
w
∈
V
(
Q
)
). We consider the following two subsubcases.
Case 3.3.1. If
W
2
contains vertex
v
3
, we may uniquely decompose
W
2
into two sections
W
21
W
22
, where
W
21
is the longest
(
v
1
,
v
3
)
section of
W
2
and
W
22
is the remaining shortest
(
v
3
,
v
2
)
section of
W
2
.
Let
η
(
W
)
=
η
(
W
1
)
η
(
W
21
)
η
(
W
22
)
η
(
W
3
)
η
(
W
4
)
, where
η
(
W
1
)
=
W
1
,
η
(
W
21
)
=
W
21
,
η
(
W
3
)
=
(
W
3

W
3
∩
(
S
m
3
∪
Q
)
)
∪
ξ
(
W
3
∩
(
S
m
3
∪
Q
)
)
,
η
(
W
22
)
is a
(
v
3
,
v
1
)
walk obtained from
W
22
replacing
v
1
by
v
2
and
v
2
by
v
1
, and
η
(
W
4
)
is a
(
v
1
,
v
3
)
(or
(
v
1
,
w
)
)walk obtained from
W
4
replacing
v
1
by
v
2
and
v
2
by
v
1
.
Case 3.3.2. If
W
2
does not contain vertex
v
3
, let
η
(
W
)
=
η
(
W
1
)
η
(
W
2
)
η
(
W
3
)
η
(
W
4
)
, where
η
(
W
1
)
=
W
1
,
η
(
W
3
)
=
(
W
3

W
3
∩
(
S
m
3
∪
Q
)
)
∪
ξ
(
W
3
∩
(
S
m
3
∪
Q
)
)
,
η
(
W
2
)
is a
(
v
1
,
v
1
)
walk obtained from
W
2
replacing its last two vertices
v
1
v
2
by
v
1
, and
η
(
W
4
)
is a
(
v
1
,
v
3
)
(or
(
v
1
,
w
)
)walk obtained from
W
4
replacing its first vertex
v
2
by
v
1
v
2
.
Case 3.4. If
W
1
does not pass edge
v
1
v
t
and
W
3
passes edge
v
1
v
t
, we may anew decompose
W
into four sections
W
1
W
2
W
3
W
4
, where
W
1
is the longest
(
v
3
,
v
2
)
(or
(
w
,
v
2
)
)section of
W
(which do not contain vertices
u
0
,
…
,
u
t
′
′
and must contain vertex
v
3
), the second
W
2
is the internal longest
(
u
0
,
u
t
′
′
)
section of
W
(for which the internal vertices are some possible vertices in
V
(
G
1
)
),
W
3
is the third
(
v
2
,
v
1
)
section of
W
(for which the internal vertices, if exist, are only possible
v
1
,
v
2
,
v
3
,
w
∈
V
(
Q
)
), and the last
W
4
is the longest
(
v
t
,
v
3
)
(or
(
v
t
,
w
)
)section of
W
. We have two subsubcases.
Case 3.4.1. If
W
3
contains vertex
v
3
, we may uniquely decompose it into two sections
W
31
W
32
, where
W
31
is the shortest
(
v
2
,
v
3
)
section of
W
3
and
W
32
is the remaining longest
(
v
3
,
v
1
)
section of
W
3
.
Let
η
(
W
)
=
η
(
W
1
)
η
(
W
2
)
η
(
W
31
)
η
(
W
32
)
η
(
W
4
)
, where
η
(
W
2
)
=
(
W
2

W
2
∩
(
S
m
3
∪
Q
)
)
∪
ξ
(
W
2
∩
(
S
m
3
∪
Q
)
)
,
η
(
W
32
)
=
W
32
,
η
(
W
4
)
=
W
4
,
η
(
W
1
)
is a
(
v
3
,
v
1
)
(or
(
w
,
v
1
)
)walk obtained from
W
1
replacing
v
1
by
v
2
and
v
2
by
v
1
, and
η
(
W
31
)
is a
(
v
1
,
v
3
)
walk obtained from
W
31
replacing
v
1
by
v
2
and
v
2
by
v
1
.
Case 3.4.2. If
W
3
does not contain vertex
v
3
, let
η
(
W
)
=
η
(
W
1
)
η
(
W
2
)
η
(
W
3
)
η
(
W
4
)
, where
η
(
W
2
)
=
(
W
2

W
2
∩
S
m
3
)
∪
ξ
(
W
2
∩
S
m
3
)
,
η
(
W
4
)
=
W
4
,
η
(
W
1
)
is a
(
v
3
,
v
1
)
(or
(
w
,
v
1
)
)walk obtained from
W
1
replacing its last vertex
v
2
by
v
2
v
1
, and
η
(
W
3
)
is a
(
v
1
,
v
1
)
walk obtained from
W
3
replacing its first two vertices
v
2
v
1
by
v
1
.
Case 4. Suppose
W
is a closed walk starting from
v
i
for
i
=
4,5
,
6
,
…
,
n
in
X
1
. For
W
∈
X
1
, we may uniquely decompose
W
into five sections
W
1
W
2
W
3
W
4
W
5
, where
W
1
is the longest
(
v
i
,
v
t
)
section of
W
(which do not contain vertices
u
0
,
…
,
u
t
′
′
),
W
2
is the second
(
v
1
,
v
2
)
section of
W
(for which the internal vertices, if exist, are only possible
v
1
,
v
2
,
v
3
,
w
∈
V
(
G
)
), the third
W
3
is the internal longest
(
u
0
,
u
t
′′
)
section of
W
(for which the internal vertices are some possible vertices in
V
(
G
1
)
), the fourth
W
4
is the longest
(
v
2
,
v
1
)
section of
W
(for which the internal vertices, if exist, are only possible
v
1
,
v
2
,
v
3
,
w
∈
V
(
G
)
), and the last
W
5
is the remaining
(
v
t
′
,
v
i
)
section of
W
. We have four subcases.
Case 4.1. If both
W
2
and
W
4
contain the vertex
v
3
, we may uniquely decompose
W
2
into two sections
W
21
W
22
and decompose
W
4
into two sections
W
41
W
42
, where
W
21
is the longest
(
v
1
,
v
3
)
section of
W
2
,
W
22
is the remaining shortest
(
v
3
,
v
2
)
of
W
2
,
W
41
is the shortest
(
v
2
,
v
3
)
section of
W
4
, and
W
42
is the remaining longest
(
v
3
,
v
1
)
section of
W
4
.
Let
η
(
W
)
=
η
(
W
1
)
η
(
W
21
)
η
(
W
22
)
η
(
W
3
)
η
(
W
41
)
η
(
W
42
)
η
(
W
5
)
, where
η
(
W
1
)
=
W
1
,
η
(
W
21
)
=
W
21
,
η
(
W
3
)
=
(
W
3

W
3
∩
(
S
m
3
∪
Q
)
)
∪
ξ
(
W
3
∩
(
S
m
3
∪
Q
)
)
,
η
(
W
42
)
=
W
42
η
(
W
5
)
=
W
5
,
η
(
W
22
)
is a
(
v
3
,
v
1
)
walk obtained from
W
22
replacing
v
1
by
v
2
and
v
2
by
v
1
, and
η
(
W
41
)
is a
(
v
1
,
v
3
)
walk obtained from
W
41
replacing
v
1
by
v
2
and
v
2
by
v
1
.
Case 4.2. If
W
2
contains the vertex
v
3
and
W
4
does not contain vertex
v
3
, let
η
(
W
)
=
η
(
W
1
)
η
(
W
2
)
η
(
W
3
)
η
(
W
4
)
η
(
W
5
)
, where
η
(
W
1
)
=
W
1
,
η
(
W
3
)
=
(
W
3

W
3
∩
(
S
m
3
∪
Q
)
)
∪
ξ
(
W
3
∩
(
S
m
3
∪
Q
)
)
,
η
(
W
5
)
=
W
5
,
η
(
W
2
)
is a
(
v
1
,
v
1
)
walk obtained from
W
2
replacing its last vertex
v
2
by
v
2
v
1
, and
η
(
W
4
)
is a
(
v
1
,
v
1
)
walk obtained from
W
4
replacing its first two vertices
v
2
v
1
by
v
1
.
Case 4.3. If
W
2
does not contain the vertex
v
3
, let
η
(
W
)
=
η
(
W
1
)
η
(
W
2
)
η
(
W
3
)
η
(
W
4
)
η
(
W
5
)
, where
η
(
W
1
)
=
W
1
,
η
(
W
3
)
=
(
W
3

W
3
∩
(
S
m
3
∪
Q
)
)
∪
ξ
(
W
3
∩
(
S
m
3
∪
Q
)
)
,
η
(
W
5
)
=
W
5
,
η
(
W
2
)
is a
(
v
1
,
v
1
)
walk obtained from
W
2
by replacing its last two vertices
v
1
v
2
by
v
1
, and
η
(
W
4
)
is a
(
v
1
,
v
1
)
walk obtained from
W
4
by replacing its first vertex
v
2
by
v
1
v
2
.
For example,
(8)
η
1
(
u
0
u
1
⋯
u
r
v
2
v
3
w
1
⋯
w
t
v
3
v
1
v
2
u
1
′
⋯
hhhh
u
s
′
v
2
v
1
v
3
v
2
u
1
′′
⋯
u
t
′′
u
0
)
=
u
0
u
1
⋯
u
r
v
1
v
3
w
1
⋯
w
t
v
3
v
2
v
1
u
1
′
⋯
u
s
′
v
1
v
2
v
3
v
1
u
1
′′
⋯
u
t
′′
u
0
,
η
1
(
u
0
u
1
⋯
u
r
v
2
v
3
w
1
⋯
w
t
v
3
v
1
v
4
v
5
v
1
v
2
u
1
′
⋯
hhhh
u
s
′
v
2
v
3
v
5
v
4
v
1
v
3
v
2
u
1
′′
⋯
u
t
′′
u
0
)
=
u
0
u
1
⋯
u
r
v
1
v
2
v
3
w
1
⋯
w
t
v
3
v
1
v
4
v
5
v
1
u
1
′
⋯
u
s
′
v
1
v
3
v
5
v
4
v
1
v
3
v
1
u
1
′′
⋯
u
t
′′
u
0
,
η
1
(
v
3
w
1
⋯
w
t
v
3
v
2
u
1
⋯
u
r
v
2
v
1
v
5
v
4
v
1
v
2
u
1
′
⋯
u
s
′
v
2
v
1
v
3
)
=
v
3
w
1
⋯
w
t
v
3
v
1
u
1
⋯
u
r
v
1
v
5
v
4
v
1
v
2
v
1
u
1
′
⋯
u
s
′
v
1
v
2
v
3
,
η
1
(
v
3
v
2
v
1
v
7
v
6
v
1
v
2
v
1
v
2
u
1
⋯
u
r
v
2
v
1
v
4
v
5
v
1
v
3
v
2
u
1
′
⋯
u
s
′
v
2
v
1
v
7
v
6
v
1
v
2
v
3
)
=
v
3
v
2
v
1
v
7
v
6
v
1
v
2
v
1
u
1
⋯
u
r
v
1
v
4
v
5
v
1
v
3
v
2
v
1
u
1
′
⋯
u
s
′
v
1
v
2
v
1
v
7
v
6
v
1
v
2
v
3
,
η
1
(
v
1
v
4
v
5
v
1
v
3
v
2
u
1
⋯
u
r
v
2
v
3
v
2
v
1
v
4
v
5
v
1
v
3
v
2
u
1
′
⋯
u
s
′
v
2
v
1
)
=
v
2
v
3
v
1
v
5
v
4
v
1
u
1
⋯
u
r
v
1
v
3
v
2
v
1
v
4
v
5
v
1
v
3
v
1
u
1
′
⋯
u
s
′
v
1
v
2
,
η
1
(
v
4
v
5
v
1
v
3
v
2
u
1
⋯
u
r
v
2
v
3
v
2
v
1
v
4
v
5
v
1
v
3
v
2
u
1
′
⋯
u
s
′
v
2
v
3
v
2
v
1
v
5
v
4
)
=
v
4
v
5
v
1
v
3
v
1
u
1
⋯
u
r
v
1
v
3
v
2
v
1
v
4
v
5
v
1
v
3
v
1
u
1
′
⋯
u
s
′
v
1
v
3
v
2
v
1
v
5
v
4
,
where
u
0
,
u
1
,
…
,
u
r
,
u
1
′
,
…
,
u
s
′
,
u
1
′′
,
…
,
u
t
′′
are vertices in
G
and
w
1
,
…
,
w
t
are vertices in
Q
.
By Lemma 1,
ξ
is injective and not surjective. It is easily shown that
η
is also injective and not surjective. Thus

X
1

<

X
2

,
M
k
(
G
1
)
<
M
k
(
G
2
)
.