Download Reciprocal Complementary Wiener Numbers of Non

Applied Mathematics, 2016, 7, 219-226
Published Online February 2016 in SciRes. http://www.scirp.org/journal/am
http://dx.doi.org/10.4236/am.2016.73020
Reciprocal Complementary Wiener
Numbers of Non-Caterpillars
Yanli Zhu, Fuyi Wei, Feng Li
Department of Applied Mathematics, South China Agricultural University,
Guangzhou, China
Received 14 January 2016; accepted 23 February 2016; published 26 February 2016
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract
The reciprocal complementary Wiener number of a connected graph G is defined as
RCW ( G ) =
1
{u , v}⊆V (G ) d + 1 − d ( u, v | G )
∑
where V ( G ) is the vertex set. d ( u, v | G ) is the distance between vertices u and v, and d
is the diameter of G. A tree is known as a caterpillar if the removal of all pendant vertices
makes it as a path. Otherwise, it is called a non-caterpillar. Among all n-vertex non-caterpillars with given diameter d, we obtain the unique tree with minimum reciprocal complementary Wiener number, where 4 ≤ d ≤ n − 3 . We also determine the n-vertex non-caterpillars with the smallest, the second smallest and the third smallest reciprocal complementary Wiener numbers.
Keywords
Reciprocal Complementary Wiener Number, Wiener Number, Caterpillar
1. Introduction
The Wiener number was one of the oldest topological indices, which was introduced by Harry Wiener in 1947.
About the recent reviews on matrices and topological indices related to Wiener number, refer to [1]-[4]. The
RCW number is one of the hotest additions in the family of such descriptors. The notion of RCW number was
first put forward by Ivanciuc and its applications were discussed in [5]-[8].
How to cite this paper: Zhu, Y.L., Wei, F.Y. and Li, F. (2016) Reciprocal Complementary Wiener Numbers of Non-Caterpillars.
Applied Mathematics, 7, 219-226. http://dx.doi.org/10.4236/am.2016.73020
Y. L. Zhu et al.
Let G be a simple connected graph with vertex set V ( G ) . For two vertices u , v ∈ V ( G ) , let d ( u , v | G )
denote the distance between u and v in G. Then, the RCW number of G is defined by
RCW ( G ) =
1
1
d
d
+
−
( u, v | G )
{u ,v}⊆V ( G )
∑
where d is the diameter and the summation goes over all unordered pairs of distinct vertices of G. Some
properties of the RCW number have been obtained in [9] [10].
A tree is called a caterpillar if the removal of all pendant vertices makes it as a path. Otherwise, it is called a
non-caterpillar.
For integers n and d satisfying 4 ≤ d ≤ n − 3 , let N n ,d ,i be the tree obtained from the path Pd +1 labelled as
d 
v0 , v1 , , vd by attaching the path P2 and n − d − 3 pendant vertices to vertex vi for 2 ≤ i ≤   (see
2
Figure 1). Let N n ,d = N  d  .
n ,d ,  
2
In this paper, we show that among all n-vertex non-caterpillars with given diameter d, N n ,d is the unique
tree with minimum RCW number where 4 ≤ d ≤ n − 3 . Furthermore, we determine the non-caterpillars with the
smallest, the second smallest and the third smallest RCW numbers.
2. RCW Numbers of Non-Caterpillars
All n-vertex trees with diameter 2, 3, n − 2 and n − 1 are caterpillars. Let n and d be integers with n ≥ 7 and
4 ≤ d ≤ n − 3 . Let NC ( n, d ) be the class of non-caterpillars with n vertices and diameter d. Let  ( n, d ) be
t
the class of non-caterpillars obtained by attaching the stars sn1 , , snt at their centers and s = n − d − 1 − ∑ni
i =1
pendant vertices to one center (fixed if it is bicentral) of the path Pd +1 , where t ≥ 1 , s ≥ 0 and ni ≥ 2 for
i = 1, 2, , t (see Figure 2). Recall that N n,d = N  d  . Obviously, N n,d ∈  ( n, d ) ⊆ NC ( n, d ) and
n ,d ,  
2
 ( n, n − 3) =
{ N n , n −3 } .
Let T be a tree. For u ∈ V (T ) and A ⊆ V (T ) , let δ T ( u ) be the degree of u in T and dT ( u , A ) be the
sum of all distances from u to the vertices in A, i.e., dT ( u , A ) = ∑dT ( u , v ) . Here and in the following dT ( u , v )
v∈A
denotes the distance between vertices u and v in T.
Lemma 1 Let T be a tree with minimum RCW number in NC ( n, d ) , where 4 ≤ d ≤ n − 3 . Then,
T ∈  ( n, d ) .
Proof. Suppose that T ∈ NC ( n, d ) \  ( n, d ) . Let P (T ) = v0 v1  vd be a diametral path of T. If d is odd,




we require that δ T  v d   ≥ δ T  v d   . Then at least one of v2 , , vd −2 has degree at least three. There are two
  
  
 2 
 2 
cases.
Case 1. One of v2 , , vd −2 different from v d  has degree at least three. Let w1 , w2 , , wk be all the
 
neighbors
2
outside P (T ) except those of v d  , where wi is a neighbor of vi′ ∈ V ( P (T ) ) . Let Ti be the subtree of
2
 
T − vi′ containing wi . T
∗
be the tree formed from T by deleting edges wi vi′ and adding edges wi v d  for
 
2
Figure 1. The tree Nn,d,i.
220
Y. L. Zhu et al.
Figure 2. The tree NC (n,d).
all i = 1, 2, , k . Obviously, T ∗ ∈ NC ( n, d ) . Let A = V ( P (T ) ) and B = V (T ) \ A . It is easily seen that
( )
=
RCW (T ) − RCW
T∗


1
1
−


u∈B ,v∈A 
 d + 1 − dT ( u , v ) d + 1 − dT ∗ ( u , v ) 
∑
+
≥


1
1
−


{u ,v}⊆ B  d + 1 − dT ( u , v ) d + 1 − dT ∗ ( u , v ) 
∑

1

1
∑ ∑  d + 1 − d ( u, v ) − d + 1 − d ( u, v ) 
u∈V (T ) v∈A 

T

T
1≤i ≤ k
+
∗
i


1
1
−


u∈V (Ti ),v∈V (T j ) 
 d + 1 − dT ( u , v ) d + 1 − dT ∗ ( u , v ) 
∑
1≤i < j ≤ k




d 
with equality if and only if δ T  v d   = 2 . Since i′ ≠   , dT ( u , vi′ ) = d ∗  u , v d   for u ∈ V (Ti ) and
T 
  

2
 2 
  2  
vi′ ∈ A with 1 ≤ i ≤ k . We get
1
1
=∑
∑
v∈A d + 1 − dT ( u , v )
v∈A d + 1 − dT ( u , vi′ ) − dT ( vi′ , v )
i′
1
d −i ′
1
∑ d + 1 − d ( u, v ) − k + ∑ d + 1 − d ( u, v ) − k
k 1=
k 1
=
T
i′
T
i′
=
d 
2
 
d 
 
2
1
1
≥∑
+∑




k 1=
k 1
=
d + 1 − dT ∗  u , v d   − k
d + 1 − d T ∗  u , v d   − k
  
  
 2 
 2 
1
1
=∑
=∑
.
+
−
d
1
d



 v∈A
( u, v )
v∈A
T∗
d + 1 − dT ∗  u , v d   − dT ∗  v d  , v 
  
   
 2 
 2 
Then

1

1
∑ ∑  d + 1 − d ( u, v ) − d + 1 − d ( u, v ) 
u∈V (T ) v∈A 

T
T

1≤i ≤ k
=
∗
i

1

1
∑  ∑ d + 1 − d ( u, v ) − ∑ d + 1 − d ( u, v )  ≥ 0
u∈V (T ) 
v∈A
 v∈A
T
T

1≤i ≤ k
∗
i


d 
with equality if and only if i′ =   (which is only possible for odd number d). But δ T  v d   ≥ δ T


2
  2  
221


 v d   , and


  2  
Y. L. Zhu et al.




d 
thus if δ T  v d   = 2 then δ T  v d   = 2 . So i′ ≠   for i = 1, 2, , k . Thus
  
  
2
 2 
 2 
( )
RCW (T ) − RCW T ∗
>


1
1
−

 ≥ 0,
u∈V (Ti ),v∈V (T j ) 
 d + 1 − dT ( u , v ) d + 1 − dT ∗ ( u , v ) 
∑
1≤i < j ≤ k
since dT ( u , v ) ≥ dT ∗ ( u , v ) for u ∈ V (Ti ) , v ∈ V (T j )
This is a contradiction.
(1 ≤ i <
j ≤ k ) . It follows that RCW (T ) > RCW (T ∗ ) .


d 
with i 2, 3, , d − 2 and i ≠   has degree two. Obviously, δ T  v d   ≥ 3 . Let
Case 2. Any verter vi =
  
2
 2 




xyz  v d  be the (unique) path from x to v d  in T such that dT  x, v d   = max u∈V (T )\V ( P(T ) ) dT  u , v d   . Since
  
  
2
2
 
 
 2 
 2 


T ∈/  ( n, d ) , we have dT  x, v d   ≥ 3 . Let x1 , , xr , z be the neighbors of y in T, where x1 = x and r ≥ 1 .
  
 2 
∗∗
Let T be the tree obtained from T by deleting edges yxi and adding edges zxi for all i = 1, 2, , r . Then
dT ( u , v ) dT ∗∗ ( u , v ) + 1 for
T ∗∗ ∈ NC ( n, d ) . Let N y = { x1 , , xr } , C = V (T ) \ N y . Since =
u ∈ N y , v ∈ C \ { y, z} , we get
( )
RCW (T ) − RCW T ∗∗


1
1
−


u∈N y ,v∈C 
 d + 1 − dT ( u , v ) d + 1 − dT ∗ ( u , v ) 
1
r −1
r −1
=
+
+
∑
d + 1 − 1 d + 1 − 2 u∈N y ,v∈C \{ y , z} d + 1 − dT ( u , v )
∑
=
−
1
r −1
r −1
−
−
∑
d + 1 − 1 d + 1 − 2 u∈N y ,v∈C \{ y , z} d + 1 − dT ∗ ( u , v )
> 0.
This is a contradiction.
By combining Cases 1 and 2, we find that T ∈ NC ( n, d ) \  ( n, d ) is impossible. The result follows.
Lemma 2 Let T ∈  ( n, d ) with 4 ≤ d ≤ n − 3 . Then
RCW (T ) ≥ RCW ( N n ,d ) ,
with equality if and only if T = N n ,d .
Proof. Let T be a tree with the minimum RCW number in  ( n, d ) . Let P (T ) = v0 v1  vd be a diametral
path of T.
Suppose that there is a vertex u ∈ V (T ) \ V ( P (T ) ) with δ T ( u ) ≥ 3. Let u1 , u2 , , us be the neighbors of u
different from v d  in T, where s ≥ 2 . Clearly, ui are pendant vertices for i = 1, 2, , s . Let T ′ be the tree
2
 deleting edges uu
′
obtained from T by
i and adding edges v d  ui for i = 2, , s . Obviously, T ∈  ( n, d ) .
2
 


dT ′ ( u , v ) + 1 for
Let N u = {u2 , , us } , D = V (T ) \ N u , and K = u , u1 , v d  , v d   . Since d=
T ( u, v )
2
 2  −1 

 
  
u ∈ N u , v ∈ D \ K , we get
222
Y. L. Zhu et al.
RCW (T ) − RCW (T ′ )


1
1
−


u∈Nu ,v∈D 
 d + 1 − dT ( u , v ) d + 1 − dT ∗ ( u , v ) 
∑
=
=
1
s −1
s −1
s −1
s −1
+
+
+
+ ∑
d + 1 − 1 d + 1 − 2 d + 1 − 2 d + 1 − 3 u∈Nu ,v∈D \ K d + 1 − dT ( u , v )
−
1
s −1
s −1
s −1
s −1
−
−
−
− ∑
d + 1 − 2 d + 1 − 1 d + 1 − 3 d + 1 − 2 u∈Nu ,v∈D \ K d + 1 − dT ′ ( u , v )
> 0,
and then RCW (T ) > RCW (T ′ ) , this is a contradiction. Thus any vertex of T outside P (T ) has degree at
most two.
Suppose that there are at least two vertices of T outside P (T ) with degree two. Let y ∈ V (T ) \ V ( P (T ) )
with δ T ( y ) = 2 and let x be the neighbor of y which is different from v d  in T. Let T ′′ be the tree formed
2
 


from T by deleting edge yx and adding edge v d  x . Obviously, T ′′ ∈  ( n, d ) . Let F =  x, y, v d   . Since
2 
2
 
 

RCW (T ) > RCW (T ′′ ) and d=
dT ′′ ( x, v ) + 1 for v ∈ V (T ) \ F , we get
T ( x, v )
RCW (T ) − RCW (T ′′ )
=
1
1
1
+
+ ∑
d + 1 − 1 d + 1 − 2 v∈V (T )\ F d + 1 − dT ( x, v )
−
1
1
1
−
− ∑
d + 1 − 1 d + 1 − 2 v∈V (T )\ F d + 1 − dT ′′ ( x, v )
> 0.
This is a contradiction. Thus there is exactly one vertex outside P (T ) with degree two and all other vertices
of T outside P (T ) are pendant vertices. Then, T = N n ,d .
By a direct calculation, we get


n − d + 1 ( n − d − 2 )( n − d − 3) − 2
RCW  N  d   =
d+
+
 n,d ,   
2 ( d − 1)
d −2
2 

 d2

2
1
+ ( n − d − 1)  ∑
+  . where d is even;
 k =1 d − k d 




( n − d − 2 )( n − d − 3) + 4 ( n − d ) − 2
2
n−d −3
RCW  N  d   =
d+
+
+
 n,d ,   
2 ( d − 1)
d −2
d −3
2 

 d 2−1

2
1

+ ( n − d − 1)  ∑
+  . where d is odd.
 k =1 d − k d 


Combining Lemmas 1 and 2, we get
Theorem 1 Let T ∈ NC ( n, d ) , and 4 ≤ d ≤ n − 3. Then
RCW (T ) ≥ RCW ( N n ,d )
with equality if and only if T = N n ,d .
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Y. L. Zhu et al.




.
Lemma 3 For 4 ≤ d ≤ n − 3 , there is RCW  N  d   > RCW  N
d +1 

 n ,d ,   
 n ,d +1,  
2 
 2  


Proof. If d is even, then




RCW  N d  − RCW  N
d 
n ,d ,
n ,d +1, 
2 
2 


d
−1


2
2  5 ( n − d − 1) n − d − 2 4 ( n − d − 2 ) + 1 n − d − 1
=  −1 + ∑
−
−
+
+
d
d +1
d
d −2
k =1 d − k 



n − d − 3 ( n − d − 2 )( n − d − 3) ( n − d − 3)( n − d − 4 )
−
+
−
d −1
2 ( d − 1)
2d
d
−1


2
2 

>  −1 + ∑
 > 0.
k =1 d − k 



If d is odd, then




RCW  N d −1  − RCW  N
d +1 

n ,d ,
n ,d +1,


2 
2 
d −1


2
2  n − d − 3 n − d − 4 ( n − d − 2 )( n − d − 3)

=  −1 + ∑
+
+ d −2 −
2 ( d − 1)
d
k =1 d − k 



( n − d − 3)( n − d − 4 ) + n − d − 1 − n − d − 2 + 2
−
2d
d −1
d +1
d −3
d −1


2
2 
>  −1 + ∑
 > 0.
k =1 d − k 



The result follows.
Theorem 2 For n ≥ 9 , there is






 < RCW  N
 < RCW  N
.
RCW  N
n −3 
n −5 
n−4 



 n ,n−3,  
 n ,n−3,  
 n ,n−4,  
 2  
 2  
 2  





 for any n-vertex non-caterpillar T different from
RCW (T ) > RCW  N
4
−
n
 n ,n−4,  
 2 

, N
.
 n−4 
 n −5 
And
N
n ,n −3, 

 2 
n ,n − 4, 

 2 
N
 n −3 
n ,n −3, 

 2 
,
Proof. Let T ∈ NC ( n, d ) , where 4 ≤ d ≤ n − 3 . If d= n − 3 , then T is a non-caterpillar N n ,n−3,i where
 n − 3
1≤ i ≤ 
. It follows that
 2 
RCW ( N n ,n−3,i ) = n − 3 +
i +3
n −i
2
1
2
2
1
1
,
+
+∑
+∑
+
+
n−3 =
n − 4 k 4=
n − k k 5 n − k n − i − 4 i −1
 n − 3
and hence RCW ( N n ,n−3,i ) is monotonically decreasing for 1 ≤ i ≤ 
. This implies
 2 
224




 < RCW  N
 <  < RCW ( N n ,n−3,2 ) .
RCW  N
 n −3 
 n −5 
 n ,n−3,  
 n ,n−3,  
 2 
 2 


Y. L. Zhu et al.
Now suppose that d ≤ n − 4 . By Theorem 1 and Lemma 3, there is



RCW (T ) ≥ RCW  N
n−4 

 n ,n−4,  
2




where equality holds if and only if T ≅ N
 n−4 
n ,n − 4, 

 2 
. We need only to show






.
<
RCW  N
RCW
N
 n ,n−3, n−5  
 n ,n−4, n−4  
 2  
 2  


n − 5 n − 5
Case 1. n is odd.=
Let i =
and n ≥ 9 . Then there is
2
 2 




RCW  N
n −5 
 − RCW  N n ,n−3, n−5  =
n ,n − 4,
2 
2 


n +5


 −1 + 2 2  + 2 + 1 > 0.
∑ n−k  n−5 n−6

k =6




n − 5 n − 6
Case 2. n is even.=
Let i =
. Then there is
2
 2 




−
RCW  N
RCW
N

n−4 
n
−
6

 n ,n−3, 
 n ,n−4, 2 

2 
n+ 4


 −1 + 2 2  + 2 + 1 + 2 + 4 − 2 − 2
=
∑ n − k  n −5 n −6 n − 2 n − 4 n −3 n −8

k =6




n+ 4


2
2 
5
2

>  −1 + ∑
+
−

n −5 n −8
k =6 n − k 



≥ 0.
Thus, the proof is finished.
References
[1]
Ivanciuc, O. (2003) Graph Theory in Chemistry. In: Gasteiger, J., Ed., Handbook of Chemoinformatics, Wiley-VCH.
Weinheim, 103-138. http://dx.doi.org/10.1002/9783527618279.ch6
[2]
Ivanciuc, O. (2003) Topological Indices. In: Gasteiger, J., Ed., Handbook of Chemoinformatics, Wiley-VCH, Weinheim, 981-1003. http://dx.doi.org/10.1002/9783527618279.ch36
[3]
Janežić, D., Miličević, A., Nikolić, S. and Trinajstić, N. (2007) Graph-Theoretical Matrices in Chemistry. University of
Kragujevac, Kragujevac.
[4]
Luo, W. and Zhou, B. (2009) On Ordinary and Reverse Wiener Indices of Non-Caterpillars. Mathematical and Computer Modelling, 50, 188-193. http://dx.doi.org/10.1016/j.mcm.2009.02.010
[5]
Ivanciuc, O. (2000) QSAR Comparative Study of Wiener Descriptors for Weighted Molecular Graphs. Journal of
Chemical Information and Modeling, 40, 1412-1422. http://dx.doi.org/10.1021/ci000068y
[6]
Ivanciuc, O., Ivanciuc, T. and Balaban, A.T. (2000) The Complementary Distance Matrix, a New Molecular Graph
Metric. ACH-Models in Chemistry, 137, 57-82.
[7]
Ivanciuc, O., Ivanciuc, T. and Balaban, A.T. (2002) Quantitative Structure-Property Relationship Evaluation of Struc-
225
Y. L. Zhu et al.
tural Descriptors Derived from the Distance and Reverse Wiener Matrices. Internet Electron. Journal of ComputerAided Molecular Design, 1, 467-487.
[8]
Ivanciuc, O., Ivanciuc, T. and Balaban, A.T. (1999) Vertex- and Edge-Weighted Molecular Graphs and Derived Structural Descriptors. In: Devillers, J. and Balaban, A.T., Eds., Topological Indices and Related Descriptors in QSAR and
QSPR, Gordon and Breach, Amsterdam, 169-220.
[9]
Cai, X. and Zhou, B. (2009) Reciprocal Complementary Wiener Numbers of Trees, Unicyclic Graphs and Bicyclic
Graphs. Discrete Applied Mathematics, 157, 3046-3054. http://dx.doi.org/10.1016/j.dam.2009.05.001
[10] Zhou, B., Cai, X. and Trinajstić, N. (2009) On Reciprocal Complementary Wiener Number. Discrete Applied Mathematics, 157, 1628-1633. http://dx.doi.org/10.1016/j.dam.2008.09.010
226