J. X. Liu

6

12

21

12

121 1

1

221212 12

2

12 1212 12

222

1

2222 2

()1

11

22 2

10

11

uNx x

vNx x

HGHG xxdx dxdx dudx du

dx

dxdvdxdvdx dxdx dxdx dx

dx dx dxdx dxdx dxdx dx

,

where denotes the vertex set adjoining to

i

Nx i

for .

1, 2i

If 12

x is a leaf of , i.e., at least one of G

12

,

x

has degree one, we can see that it is a local maximum

edge. Thus, by Lemma 2.3,

Corollary 2.4 If 12

x is a leaf in graph G, then

12 0.HGHG xx

Theorem 2.5 Let T be a tree with order

4n

diameter , then

DT

51

,

6223 1

HT

n

HT DTDT n

1

where equalities hold if and only if

is a path .

n

P

Proof. If is a path, we have

T

1

26

n

HT and

1DT n. It is obvious that both equalities hold.

Now we assume that

is not a path, then

and there are at least three pendent

vertices in

2DT n

. Assume 01

Puu u be the longest

path in

. Then at least one pendent vertex, say 1, is

not contained in

v

. Now we start an operation on

,

i.e., we continually delete pendent vertices which are not

contained in

until the resulting tree is

. Assume

1 are the vertices in the order they were deleted,

we have

,,

k

vv

1

1

1

23

k

i

i

D

HTHT vHTvHP

by Corollary 2.4 and

1

1

k

i

i

DTDT vDTvD

. Thus, we

have

>

51515

>

62 6262

TDTHPDP

Dn

n

and

15 1

2626

>= >

1

Dn

HT HP

DT DPDn

.

This result seems true for any connected graph with

order and we propose as a conjecture as follows:

n

Conjecture 2.6 Let be a connected graph with

order

G

4n diameter , then

DG

51

,

6223 1

HG

n

HG DGDG n

1

.

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