The maximum Wiener polarity index of unicyclic graphs

Applied Mathematics and Computation 218 (2012) 10149–10157

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The maximum Wiener polarity index of unicyclic graphs q Huoquan Hou, Bolian Liu ⇑, Yufei Huang College of Mathematical Science, South China Normal University, Guangzhou 510631, PR China

a r t i c l e

i n f o

Keywords: Wiener polarity index Unicyclic graph Distance Extremal

a b s t r a c t The Wiener polarity index of a graph G is the number of unordered pairs of vertices u; v such that the distance between u and v is 3. In this paper, we obtain a upper bound for the Wiener polarity index of unicyclic chemical graphs. Moreover, the maximum Wiener polarity index of unicyclic graphs is determined, and the corresponding extremal graphs are characterized. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Let G ¼ ðV; EÞ be a connected simple graph with jVj ¼ n and jEj ¼ m. We refer to G as an ðn; mÞ-graph. If m ¼ n þ c  1, then G is called a c-cyclic graph. Especially, if c ¼ 1, then G is called a unicyclic graph. Let N G ðuÞ be the neighbor vertex set of u, then dG ðuÞ ¼ jN G ðuÞj is called the degree of u. For i ¼ 2; 3; . . . ; N iG ðuÞ ¼ fv 2 VðGÞjdG ðu; v Þ ¼ ig is called the ith neighbor vertex set of u. If dG ðv Þ ¼ 1, then we call v a pendent vertex of G. The girth of a unicyclic graph G is defined to be the length of the cycle, denoted by gðGÞ. As usual, let K 1;n1 ; C n and P n be the star, cycle and path of order n, respectively. A hanging tree on vertex v in G, denote by T½v , is a rooted tree whose root is the vertex v. The distance dG ðu; v Þ between the vertices u and v of G is defined as the length of a shortest path connecting u and v. Let cðG; kÞ denote the number of unordered vertices pairs of G, each of whose distance is equal to k. The Wiener polarity index, denoted by W P ðGÞ, is defined to be the number of unordered vertices pairs of distance 3, i.e., W P ðGÞ ¼ cðG; 3Þ. The Wiener polarity index for the quantity defined in the equation above is introduced by Wiener [1] for acyclic molecules in a slightly P different-yet equivalent-manner. Moreover, Wiener [1] used a linear formula for the Wiener index W :¼ fu;v g # V dG ðu; v Þ and the Wiener polarity index W P to calculate the boiling points tB of the paraffins, i.e.,

tB ¼ aW þ bW P þ c; where a; b and c are constants for a given isomeric group. However, it seems that less attention has been paid for the Wiener polarity index afterwards. In 1998, by using the Wiener polarity index, Lukovits and Linert [2] demonstrated quantitative structure–property relationships in a series of acyclic and cycle-containing hydrocarbons. Besides, a physico-chemical interpretation of W P ðGÞ was found by Hosoya [3]. Recently, Du et al. [6] obtained the smallest and largest Wiener polarity indices together with the corresponding graphs among all trees on n vertices, respectively. Liu and Liu [4] described the relations between Wiener polarity index and Zagreb index, Wiener index, hyper-Wiener index, and determined the first two smallest Wiener polarity indices among all unicyclic graphs of order n. Moreover, Deng [5] characterized the extremal Wiener polarity indices among all chemical trees of order n.

q

This work is supported by NNSF of China (No. 11071088).

⇑ Corresponding author.

E-mail address: [email protected] (B. Liu). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.03.090

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P Observe that W P ðGÞ ¼ ti¼1 W P ðGi Þ if G1 ; . . . ; Gt are the connected components of a graph G. Therefore, it will suffice to consider the Wiener polarity index of connected graphs. In Section 2, we obtain a upper bound for the Wiener polarity index of unicyclic chemical graphs. In Section 3, the maximum Wiener polarity index of unicyclic graphs is determined, and the corresponding extremal graphs are characterized. 2. A upper bound for the Wiener polarity index of unicyclic chemical graphs Lemma 2.1 [6] Let T ¼ ðV; EÞ be a tree. Then

W P ðTÞ ¼

P uv 2E

ðdT ðuÞ  1ÞðdT ðv Þ  1Þ:

Lemma 2.2 [4]Let U ¼ ðV; EÞ be a unicyclic graph. Let C denote the unique cycle of U. If gðUÞ ¼ 3 with VðCÞ ¼ fv 1 ; v 2 ; v 3 g, then

W P ðUÞ ¼

P uv 2E

ðdU ðuÞ  1ÞðdU ðv Þ  1Þ þ 9  2dU ðv 1 Þ  2dU ðv 2 Þ  2dU ðv 3 Þ;

If gðUÞ ¼ 4 with VðCÞ ¼ fv 1 ; v 2 ; v 3 ; v 4 g, then

W P ðUÞ ¼

P uv 2E

ðdU ðuÞ  1ÞðdU ðv Þ  1Þ þ 4  dU ðv 1 Þ  dU ðv 2 Þ  dU ðv 3 Þ  dU ðv 4 Þ;

Moreover, if gðUÞ P 5, then we have

8P > < Puv 2E ðdU ðuÞ  1ÞðdU ðv Þ  1Þ  5; if gðUÞ ¼ 5; if gðUÞ ¼ 6; W P ðUÞ ¼ uv 2E ðdU ðuÞ  1ÞðdU ðv Þ  1Þ  3; > :P ðd ðuÞ  1Þðd ð v Þ  1Þ; if gðUÞ P 7: U U uv 2E Remark 1. The first Zagreb index M 1 ðGÞ and the second Zagreb index M2 ðGÞ (see [7]) of a graph G ¼ ðV; EÞ are defined as

M1 ðGÞ ¼

P v 2V

dG ðv Þ2

and M 2 ðGÞ ¼

P

dG ðuÞdG ðv Þ:

uv 2E

It can be seen that (also see [4])

P uv 2E

ðdG ðuÞ  1ÞðdG ðv Þ  1Þ ¼ M 2 ðGÞ  M1 ðGÞ þ jEj:

Combining the above equation with Lemmas 2.1 and 2.2, we conclude that the Wiener polarity index of trees and unicyclic graphs is directly connected with the first and the second Zagreb indices. However, the corresponding extremal unicyclic graphs of the maximum second Zagreb index are not always the corresponding extremal unicyclic graphs of the minimum first Zagreb index (see e.g. [8,9]), so it is worth to study the maximum Wiener polarity index of unicyclic graphs and the corresponding extremal unicyclic graphs. Lemma 2.3 [5]Let T be a chemical tree with n P 7 vertices. Then

W P ðTÞ 6 3n  15: Now we study the upper bound for the Wiener polarity index of unicyclic chemical graphs. Lemma 2.4. Let U be a unicyclic chemical graph with n vertices. (1) If gðUÞ ¼ 3, then W P ðUÞ 6 3n  3. (2) If gðUÞ ¼ 4, then W P ðUÞ 6 3n  4.

Proof. We only prove the first assertion, and the second assertion can be proved analogously. For a unicyclic chemical graph U with gðUÞ ¼ 3, let v ; w; u be the three vertices on the unique cycle C of U, and let dU ðv Þ ¼ t1 þ 2; dU ðuÞ ¼ t2 þ 2; dU ðwÞ ¼ t3 þ 2, N U ðv Þ ¼ fw; u; v 1 ; . . . ; v t1 g, N U ðuÞ ¼ fw; v ; u1 ; . . . ; ut2 g, N U ðwÞ ¼ fv ; u; w1 ; . . . ; wt3 g, where 0 6 t 1 ; t 2 ; t 3 6 2. Note that we can get a tree T by deleting an edge belonging to EðCÞ. Without loss of generality, suppose deleting the edge wu, then by Lemmas 2.1 and 2.2, we have

W P ðUÞ  W P ðTÞ ¼ dU ðu1 Þ þ    þ dU ðut2 Þ þ dU ðw1 Þ þ    þ dU ðwt3 Þ þ t 2 t3  2t 2  2t3 :

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Since U is a unicyclic chemical graph, without loss of generality, we may assume that 0 6 t2 6 t 3 6 2. Then

W P ðUÞ  W P ðTÞ ¼

8 0; > > > > > > dU ðw1 Þ  2 6 2; > > > > > < dU ðw1 Þ þ dU ðw2 Þ  4 6 4;

if t2 ¼ t 3 ¼ 0; if t2 ¼ 0; t 3 ¼ 1; if t2 ¼ 0; t 3 ¼ 2;

> > dU ðu1 Þ þ dU ðw1 Þ þ 1  2  2 6 5; if t2 ¼ 1; t3 ¼ 1; > > > > > > if t2 ¼ 1; t3 ¼ 2; dU ðu1 Þ þ dU ðw1 Þ þ dU ðw2 Þ þ 2  2  4 6 8; > > > : dU ðu1 Þ þ dU ðu2 Þ þ dU ðw1 Þ þ dU ðw2 Þ  4 6 12; if t2 ¼ 2; t3 ¼ 2:

By Lemma 2.3, we obtain that W P ðUÞ 6 W P ðTÞ þ 12 6 3n  3. h Lemma 2.5. Let U be a unicyclic chemical graph with n vertices. (1) If gðUÞ P 7, then W P ðUÞ 6 3n þ 12. (2) If gðUÞ ¼ 6, then W P ðUÞ 6 3n þ 9. (3) If gðUÞ ¼ 5, then W P ðUÞ 6 3n þ 7. Proof. We only prove the first assertion and the other assertions can be proved similarly. Let U be a unicyclic chemical graph with gðUÞ P 7. Clearly, we can get a tree T by deleting an edge uv 2 EðCÞ. By Lemma 2.2, we have

W P ðUÞ ¼ W P ðTÞ þ kl þ

k P i¼1

dU ðui Þ þ

l P

dU ðv j Þ  k  l;

j¼1

where dU ðuÞ ¼ k þ 1; dU ðv Þ ¼ l þ 1; N U ðuÞ ¼ fv ; u1 ; . . . ; uk g; N U ðv Þ ¼ fu; v 1 ; . . . ; v l g. Without loss of generality, assume that 1 6 k 6 l 6 3, then

W P ðUÞ  W P ðTÞ ¼

8 1 þ dU ðu1 Þ þ dU ðv 1 Þ  1  1 6 7; > > > > > > 2 þ dU ðu1 Þ þ dU ðv 1 Þ þ dU ðv 2 Þ  1  2 6 11; > > > > > < 3 þ dU ðu1 Þ þ dU ðv 1 Þ þ dU ðv 2 Þ þ dU ðv 3 Þ  1  3 6 15;

if k ¼ 1; l ¼ 1; if k ¼ 1; l ¼ 2; if k ¼ 1; l ¼ 3;

> > 4 þ dU ðu1 Þ þ dU ðu2 Þ þ dU ðv 1 Þ þ dU ðv 2 Þ  2  2 6 16; if k ¼ 2; l ¼ 2; > > > > > > if k ¼ 2; l ¼ 3; 6 þ dU ðu1 Þ þ dU ðu2 Þ þ dU ðv 1 Þ þ dU ðv 2 Þ þ dU ðv 3 Þ  2  3 6 21; > > > : 9 þ dU ðu1 Þ þ dU ðu2 Þ þ dU ðu3 Þ þ dU ðv 1 Þ þ dU ðv 2 Þ þ dU ðv 3 Þ  6 6 27; if k ¼ 3; l ¼ 3:

Hence by Lemma 2.3, we get W P ðUÞ 6 W P ðTÞ þ 27 6 3n þ 12. h By Lemmas 2.4 and 2.5, we have the following theorem. Theorem 2.6. Let U be a unicyclic chemical graph with n P 5 vertices. Then

W P ðUÞ 6 3n þ 12: 3. The maximum Wiener polarity index of unicyclic graphs The goal of this section is to characterize the unicyclic graphs of given order maximizing the Wiener polarity index. We first introduce some transformations on unicyclic graphs with gðUÞ P 4. Sigma: Let T½v i  denote a hanging tree on vertex v i of unicyclic graph U with gðUÞ P 4, where v i 2 VðCÞ and C ¼ v 1 v 2 . . . v g v 1 be the cycle of U. Among all hanging trees, suppose Pl ¼ v i u1 . . . ul1 ul is one of the longest path from the root v i to a leaf ul of the hang tree T½v i . If l P 2, then after deleting the edge v i u1 from U, we obtain a unicyclic graph A and a tree B such that v i 2 A and u1 2 B. Let U  denote the unicyclic graph obtained from A and B by identifying u1 and v iþ1ðmod gÞ and adding a new leaf v i w1 to v i (see Fig. 1). Lemma 3.1. Let U be a unicyclic graph with gðUÞ P 4. Let U  be the unicyclic graph obtained from U by applying transformation Sigma. If gðUÞ P 5, then

W P ðUÞ < W P ðU  Þ; If gðUÞ ¼ 4, then

W P ðUÞ 6 W P ðU  Þ:

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Fig. 1. Sigma.

Proof. Let De ðUÞ ¼ ffu; v gjdU ðu; v Þ ¼ 3 and there is a shortest path from u to v pass through the edge e = ab, where a, b, u, v 2 VðUÞg. Then

8P jDe ðUÞj; > > > Pe2E > > < e2E jDe ðUÞj  3; W P ðUÞ ¼ P > > e2E jDe ðUÞj  5; > > > :P e2E jDe ðUÞj þ 4  dU ðv 1 Þ  dU ðv 2 Þ  dU ðv 3 Þ  dU ðv 4 Þ;

if gðUÞ P 7; if gðUÞ ¼ 6; if gðUÞ ¼ 5; if gðUÞ ¼ 4:

Using the same notations in transformation Sigma, without loss of generality, suppose Pl ¼ v 1 u1 . . . ul1 ul ; N U ðu1 Þ ¼ fv 1 ; u2 ; u11 ; . . . ; u1p g and N U ðv 2 Þ ¼ fv 1 ; v 3 ; u21 ; . . . ; u2q g. It can be check directly that

jDv 1 w1 ðU  Þj ¼ 0;

jDv 1 v 2 ðU  Þj ¼ ðdU ðv 1 Þ  1ÞðdU ðv 2 Þ þ dU ðu1 Þ  1  1Þ;

jDv 2 u2 ðU  Þj ¼ ðdU ðu2 Þ  1ÞðdU ðv 2 Þ þ dU ðu1 Þ  1  1Þ; jDv 2 v 3 ðU  Þj ¼ ðdU ðv 3 Þ  1ÞðdU ðv 2 Þ þ dU ðu1 Þ  1  1Þ; jDu1i v 2 ðU  Þj ¼ ðdU ðu1i Þ  1ÞðdU ðu1 Þ  1 þ dU ðv 2 Þ  1Þ ði ¼ 0; . . . ; pÞ; jDu2j v 2 ðU  Þj ¼ ðdU ðu2j Þ  1ÞðdU ðu1 Þ  1 þ dU ðv 2 Þ  1Þ ðj ¼ 0; . . . ; qÞ; jDv 1 u1 ðUÞj ¼ ðdU ðv 1 Þ  1ÞðdU ðu1 Þ  1Þ;

jDv 1 v 2 ðUÞj ¼ ðdU ðv 1 Þ  1ÞðdU ðv 2 Þ  1Þ;

jDu1 u2 ðUÞj ¼ ðdU ðu2 Þ  1ÞðdU ðu1 Þ  1Þ;

jDv 2 v 3 ðUÞj ¼ ðdU ðv 3 Þ  1ÞðdU ðv 2 Þ  1Þ;

jDu1i u1 ðUÞj ¼ ðdU ðu1i Þ  1ÞðdU ðu1 Þ  1Þ ði ¼ 0; . . . ; pÞ; jDu2j v 2 ðUÞj ¼ ðdU ðu2j Þ  1ÞðdU ðv 2 Þ  1Þ ðj ¼ 0; . . . ; qÞ: Therefore, when gðUÞ P 5, we have

W P ðU  Þ  W P ðUÞ ¼ jDv 1 w1 ðU  Þj þ jDv 1 v 2 ðU  Þj þ jDv 2 u2 ðU  Þj þ jDv 2 v 3 ðU  Þj þ  jDv 1 v 2 ðUÞj  jDu1 u2 ðUÞj  jDv 2 v 3 ðUÞj 

p P i¼0

Xp

jDu1i u1 ðUÞj 

i¼0 q P j¼0

jDu1i v 2 ðU  Þj þ

q P j¼0

jDu2j v 2 ðU  Þj  jDv 1 u1 ðUÞj

jDu2j v 2 ðUÞj

P ðdU ðu2 Þ  1ÞðdU ðv 2 Þ  1Þ þ ðdU ðv 3 Þ  1ÞðdU ðu1 Þ  1Þ P dU ðu1 Þ  1 > 0: When gðUÞ ¼ 4, we obtain that

W P ðU  Þ  W P ðUÞ ¼

P

jDe ðU  Þj 

e2E

P

jDe ðUÞj  dU ðv 2 Þ þ dU ðv 2 Þ P dU ðu1 Þ  1  ðdU ðu1 Þ  1Þ P 0:

e2E

This completes the proof. h Let C g ¼ v 1 v 2 . . . v g v 1 be a cycle of order gðP 3Þ. Let C g ðk1 ; . . . ; kg Þ denote a caterpillar cycle, which is a unicyclic graph obtained from C g by attaching ki vertices to vertex v i , where ki P 0 for i ¼ 1; 2; . . . ; g. By applying transformation Sigma on a unicyclic graph U with girth gðP 4Þ and its resultant graphs repeatedly, we can get a caterpillar cycle C g ðk1 ; . . . ; kg Þ ultimately. And by Lemma 3.1, we have

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W P ðUÞ 6 W P ðC g ðk1 ; . . . ; kg ÞÞ;

ð1Þ

especially when g P 5, the equality holds if and only if U ffi C g ðk1 ; . . . ; kg Þ. Edge rotation: Let C g ¼ v 1 v 2 . . . v g v 1 be the cycle of C g ðk1 ; . . . ; kg Þ with g P 4. Without loss of generality, let v 1 and v 3 be the two vertices such that

dðv 1 Þ þ dðv 3 Þ ¼ maxfdðv i Þ þ dðv iþ2 Þg; where i  1; 2; . . . ; gðmod gÞ. We can get a new graph

C g ðk1 ; k2 þ ki ; k3 ; . . . ; ki1 ; 0; kiþ1 ; . . . ; kg Þ by removing ki pendants of

v i to v 2 ð4 6 i 6 gÞ (See Fig. 2).

Lemma 3.2. Let g P 4. Then we have

W P ðC g ðk1 ; . . . ; kg ÞÞ 6 W P ðC g ðk1 ; k2 þ ki ; k3 ; . . . ; ki1 ; 0; kiþ1 ; . . . ; kg ÞÞ: Proof. It is easy to obtain that

W P ðC g ðk1 ; k2 þ ki ; k3 ; . . . ; ki1 ; 0; kiþ1 ; . . . ; kg ÞÞ  W P ðC g ðk1 ; . . . ; kg ÞÞ ¼ ðk1 þ k3 Þðk2 þ ki Þ  ðk1 k2 þ k2 k3 þ ki ki1 þ ki kiþ1 Þ ¼ ðk1 þ k3 Þki  ðki1 þ kiþ1 Þki P 0: The proof of Lemma 3.2 is finished.

h

Applying Edge rotation on C g ðk1 ; . . . ; kg Þ P C g ðk1 ; k2 þ gi¼4 ki ; k3 ; 0; . . . ; 0Þ satisfying

and

its

resultant

graphs

repeatedly,

we

can

get

new

graph

   g P W P ðC g ðk1 ; . . . ; kg ÞÞ 6 W P C g k1 ; k2 þ ki ; k3 ; 0; . . . ; 0 :

ð2Þ

i¼4

Lemma 3.3. Let C g ðk1 ; k2 ; k3 ; 0; . . . ; 0Þ be a caterpillar cycle of order n, where g P 4 and ki P 0 for i ¼ 1; 2; 3. Then P3 i¼1 ki ¼ n  g, and

8 n  4; > > > < 2n  10; ng ng cd eþ W P ðC g ðk1 ; k2 ; k3 ; 0; . . . ; 0ÞÞ 6 b > 2 2 2n  9; > > : 2n  g;

with equality if and only if k2 ¼ Proof. It is obvious that

P3

i¼1 ki

bng c 2

or

if g ¼ 4; if g ¼ 5; if g ¼ 6; if g P 7;

dng e. 2

¼ n  g. Since g P 4, then

8 n  4; > > > < 2n  10; W P ðC g ðk1 ; k2 ; k3 ; 0; . . . ; 0ÞÞ ¼ k2  ðn  g  k2 Þ þ > 2n  9; > > : 2n  g; with equality if and only if k2 ¼

bng c 2

or

dng e. 2

8 n  4; > > > < 2n  10; if g ¼ 5; ng ng cd eþ 6b > 2n  9; 2 2 if g ¼ 6; > > : if g P 7; 2n  g;

if g ¼ 4;

if g ¼ 4; if g ¼ 5; if g ¼ 6; if g P 7;

h

In the following paper, let U g denote the caterpillar cycle C g ðk1 ; k2 ; k3 ; 0; . . . ; 0Þ of order n with k2 ¼ bng c or dng e, where 2 2 g ¼ 4; 5; . . ..

Fig. 2. Edge rotation.

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Lemma 3.4. Let n > g P 7. Then W P ðU g Þ > W P ðU gþ1 Þ. Proof. It is easy to verify that

( ng1 W P ðU g Þ  W P ðU gþ1 Þ ¼

 ngþ1  ðng1 Þ2 þ 1; if n  g is odd; 2 2

2 ðng Þ2 2



ng2 2



ng 2

þ 1;

if n  g is even;

( 2ðngÞ2 ¼

4 2ðngÞ 4

þ 1 > 0; if n  g is odd;

þ 1 > 0;

if n  g is even:

Hence we obtain the result as desired. h Theorem 3.5. Let U be a unicyclic graph of order nðP 4Þ with girth gðUÞ P 4. (1) (2) (3) (4)

If If If If

n ¼ 4, n ¼ 5, n ¼ 6, n ¼ 7,

then U ffi U 4 and W P ðU 4 Þ ¼ 0; then U 2 fU 4 ; U 5 g and W P ðU 5 Þ ¼ 0 < W P ðU 4 Þ ¼ 1; then W P ðUÞ 6 W P ðU 4 Þ ¼ W P ðU 6 Þ ¼ 3; then

W P ðUÞ 6 W P ðU 7 Þ ¼ 7 and the equality holds if and only if U ffi U 7 ; (5) If n ¼ 8, then

W P ðUÞ 6 W P ðU 7 Þ ¼ 9 and the equality holds if and only if U ffi U 7 ; (6) If n ¼ 9, then

W P ðUÞ 6 W P ðU 5 Þ ¼ W P ðU 7 Þ ¼ 12 and the equality holds if and only if U ffi U 7 or U 5 ; (7) If n ¼ 10, then

W P ðUÞ 6 W P ðU 5 Þ ¼ 16 and the equality holds if and only if U ffi U 5 ; (8) If n P 11, then

W P ðUÞ 6 W P ðU 5 Þ and the equality holds if and only if U ffi U 5 .

Proof. Combining Eqs. 1 and 2 and Lemmas 3.1, 3.2, 3.3, we have

W P ðUÞ 6 W P ðU gðUÞ Þ; especially when gðUÞ P 5, the equality holds if and only if U ffi U gðUÞ . Moreover, by Lemma 3.4, if gðUÞ P 8, then

W P ðUÞ 6 W P ðU gðUÞ Þ < W P ðU 7 Þ: Therefore, it will suffice to compare the values of W P ðU 4 Þ; W P ðU 5 Þ; W P ðU 6 Þ; W P ðU 7 Þ. It follows from Lemma 3.3 that

WP ðU 4 Þ ¼

 

W P ðU 5 Þ ¼  W P ðU 6 Þ ¼  W P ðU 7 Þ ¼

n4 2 n5 2 n6 2 n7 2



 n4 þ n  4; 2



 n5 þ 2ðn  5Þ; 2



 n6 þ 2ðn  6Þ þ 3; 2



 n7 þ 2ðn  7Þ þ 7: 2

By directly computing, we obtain the desired results.

h

Let C  3;n3 be the unicyclic graph obtained from the cycle C 3 ¼ v 1 v 2 v 3 and n  3 vertices by attaching the n  3 vertices to vertex v 1 , where n P 4. Let Sði; n  kÞ be the unicyclic graph of order n þ i  k obtained from C  3;i3 and n  k vertices by

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n attaching each of the n  k vertices to any pendent vertices of C  3;i3 , where i P 4 and n  k P 1. Let Si;nk;ki be a graph obtained from Sði; n  kÞ and k  i vertices by attaching each of k  i vertices to any second neighbor vertices of v 1 , where i P 4; n  k P 1 and k  i P 1. Denote the caterpillar cycle C 3 ðk1 ; k2 ; k3 Þ with jki  kj j 6 1 ði; j ¼ 1; 2; 3Þ of order n by U 3 .





nþ2 Lemma 3.6. Let U be a unicyclic graph with nðP 5Þ vertices and gðUÞ ¼ 3, let k ¼ dnþ2 2 e or b 2 cðP 4Þ, i P 4; n  k P 1 and  k  i P 1.

(1) If 5 6 n 6 8, then

W p ðUÞ 6



n2 2



 n2 ; 2 







with equality if and only if U ffi Sðk ; n  k Þ; Sni;nk ;k i . (2) If n ¼ 9; 10, then

W p ðUÞ 6



n2 2



 n2 ; 2

with equality if and only if U ffi Sðk ; n  k Þ; Sni;nk ;k i ; U 3 . (3) If n P 11, then

W p ðUÞ 6 W P ðU 3 Þ; with equality if and only if U ffi U 3 .

Proof. Let C ¼ v 1 v 2 v 3 be the cycle of U; T½v i  be a hang tree rooted at v i ði ¼ 1; 2; 3Þ. Let Pl ¼ v i u1 . . . us be one of the longest path from root v i to a leaf us in the hang tree T½v i . Denote by

U ¼



U  us1 us þ us3 us ; if s > 3; U  us1 us þ v i us ;

if s ¼ 3:

It can be seen that

8 if s > 4; > < dU ðus4 Þ  1 P 1; W P ðU  Þ  W P ðUÞ ¼ dU ðv i Þ  1 P 1; if s ¼ 4; > : dU ðv iþ1 Þ  2 þ dU ðv i1 Þ  2 P 0; if s ¼ 3;

ð3Þ

where fi  1; i; i þ 1g ¼ f1; 2; 3g. Applying such operation on U and its resultant graphs until s 6 3. Therefore, we just need to consider the unicyclic graph U with s 6 3 in the following. Without loss of generality, suppose

dU ðv 1 Þ P dU ðv 2 Þ P dU ðv 3 ÞðP 2Þ: Case 1. s ¼ 3 and dU ðv 2 Þ ¼ dU ðv 3 Þ ¼ 2. It is obvious that U ffi Sni;nk;ki , where i P 4; n  k P 1 and k  i P 1. From Equation 3, in such case we have W P ðU  Þ ¼ W P ðUÞ, and hence

W P ðUÞ ¼ W P ðSni;nk;ki Þ ¼ W P ðSðk; n  kÞÞ ¼ ðk  2Þðn  kÞ 6



n2 2



 n2 ; 2

with equality if and only if k ¼ dnþ2 e or bnþ2 c. 2 2 Case 2. s ¼ 3 and dU ðv 2 Þ > 2; dU ðv 3 Þ P 2. It can be seen from Equation 3 that W P ðU  Þ > W P ðUÞ. Thus we can get a new graph U by removing all such vertices us to v 1 such that W p ðUÞ > W p ðUÞ and the length of the longest path Pl from v i to a pendant vertex in the hang tree T½v i  of U is 2. Case 3. s ¼ 2 and dU ðv 2 Þ ¼ dU ðv 3 Þ ¼ 2. Consequently, U ffi Sðk; n  kÞ and



W P ðUÞ ¼ W P ðSðk; n  kÞÞ 6 with equality if and only if k ¼



n2 2

nþ2 2



or

 

n2 2

nþ2 2



 .

Case 4. s ¼ 2 and dU ðv 2 Þ > 2; dU ðv 3 Þ P 2. We may assume that P l ¼ v 1 u1 . . . us . Let U  ¼ U  us1 us þ v 3 us . Then

W P ðU  Þ P W P ðUÞ; with equality if and only if dU ðv 2 Þ ¼ 3. By such operation we can finally get a caterpillar cycle C 3 ðk1 ; k2 ; k3 Þ and the following assertions hold: (1) if n 6 7 and dU ðv 2 Þ ¼ 3, then U ffi U 3 ;

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H. Hou et al. / Applied Mathematics and Computation 218 (2012) 10149–10157

(2) if n > 7 and dU ðv 2 Þ ¼ 3, then U ffi C 3 ðk1 ; k2 ; k3 ÞÀU 3 and

W P ðUÞ ¼ W P ðC 3 ðk1 ; k2 ; k3 ÞÞ < W P ðU 3 Þ: (3) if dU ðv 2 Þ > 3, then by directly computing,

W P ðUÞ < W P ðC 3 ðk1 ; k2 ; k3 ÞÞ 6 W P ðU 3 Þ: Case 5. s ¼ 1. Then U ffi C 3 ðk1 ; k2 ; k3 Þ, and we have

W P ðC 3 ðk1 ; k2 ; k3 ÞÞ 6 W P ðU 3 Þ; with equality if and only if C 3 ðk1 ; k2 ; k3 Þ ffi U 3 .   Combining the above cases, it will suffice to compare the Wiener polarity indices of Sðk ; n  k Þ ðor Sni;nk ;k i Þ and U 3 :





W P ðU 3 Þ  W P ðSðk ; n  k ÞÞ ¼ W P ðU 3 Þ  W P ðSni;nk ;k i Þ ¼

¼

8 > ðn3 Þ2  3  ðn2  1Þ2 ; if n  0ðmod 6Þ; > 3 > > > > n4 n1 n4 2 n3 n1 > > > 3 3  2 þ ð 3 Þ  2 2 ; if n  1ðmod 6Þ; > > > > < n2 n5  2 þ ðn2 Þ2  ðn2 Þ2 ; if n  2ðmod 6Þ; 3 3 3 2

> n1 > 3  ðn3 Þ2  n3 ; if n  3ðmod 6Þ; > 3 2 2 > > > > 2 2 n4 n1 > >  2 þ ðn4 Þ  ðn2 Þ ; if n  4ðmod 6Þ; > 3 2 > 3 3 > > : n5 n2 2 n1  2 þ ðn2 Þ  n3 ; if n  5ðmod 6Þ: 3 3 3 2 2 pffiffiffi pffiffiffi 81 ðn  6  2 3Þðn  6 þ 2 3Þ; if n  0ðmod 6Þ; > 12 > > > pffiffiffiffiffiffi pffiffiffiffiffiffi > 1 > > ðn  6  13Þðn  6 þ 13Þ; if n  1ðmod 6Þ; > 12 > > > < 1 ðn  2Þðn  10Þ; if n  2ðmod 6Þ; 12

1 > ðn  3Þðn  9Þ; if n  3ðmod 6Þ; > > 12 > > >1 > ðn  2Þðn  10Þ; if n  4ðmod 6Þ; > > 12 > > pffiffiffiffiffiffi pffiffiffiffiffiffi : 1 ðn  6  13Þðn  6 þ 13Þ; if n  5ðmod 6Þ: 12

Hence the results follow. This completes the proof.

h

Finally, by Theorem 3.5 and Lemma 3.6, the maximum Wiener polarity index of unicyclic graphs is determined. Theorem 3.7. Let U be a unicyclic graph of order nðP 3Þ. (1) If n ¼ 3, then U ffi U 3 and W P ðU 3 Þ ¼ 0; (2) If n ¼ 4, then U 2 fU 3 ; U 4 g and W P ðU 3 Þ ¼ W P ðU 4 Þ ¼ 0; (3) If n ¼ 5, then

W P ðUÞ 6 2 and the equality holds if and only if U ffi Sð4; 1Þ; (4) If n ¼ 6, then

W P ðUÞ 6 4 and the equality holds if and only if U ffi Sð4; 2Þ; (5) If n ¼ 7, then

W P ðUÞ 6 7 and the equality holds if and only if U ffi U 7 ; (6) If n ¼ 8, then

W P ðUÞ 6 9 and the equality holds if and only if U ffi Sð5; 3Þ; Sn4;3;1 ; U 7 ; (7) If n ¼ 9, then

W P ðUÞ 6 12 and the equality holds if and only if U ffi Sð5; 4Þ; Sð6; 3Þ; Sn4;4;1 ; Sn5;3;1 ; Sn4;3;2 ; U 3 ; U 5 ; U 7 ;

H. Hou et al. / Applied Mathematics and Computation 218 (2012) 10149–10157

10157

(8) If n ¼ 10, then

W P ðUÞ 6 16 and the equality holds if and only if U ffi Sð6; 4Þ; Sn5;4;1 ; Sn4;4;2 ; U 3 ; U 5 ; (9) If n ¼ 11, then

W P ðUÞ 6 21 and the equality holds if and only if U ffi U 3 ; U 5 . (10) If n P 12, then

W P ðUÞ 6 W P ðU 3 Þ and the equality holds if and only if U ffi U 3 . Proof. The cases of n ¼ 3; 4 are obvious, and the cases of n ¼ 5; 6; 7; 8; 9; 10 can be obtained from comparing the results of Theorem 3.5 and Lemma 3.6. When n P 11, it is not difficult to obtain that

8 n4 > ðn3 Þ2  3  n6  2ðn  5Þ; if n  0ðmod > 3 2 2 > > > 2 2 > n4 n1 n4 n5 >  2 þ ð 3 Þ  ð 2 Þ  2ðn  5Þ; if n  1ðmod > 3 3 > > > < n2 n5  2 þ ðn2 Þ2  n6 n4  2ðn  5Þ; if n  2ðmod 3 3 3 2 2 W P ðU 3 Þ  W P ðU 5 Þ ¼ > 3  ðn3 Þ2  ðn5 Þ2  2ðn  5Þ; if n  3ðmod > > 3 2 > > > n4 n1 n4 2 n6 n4 >  2 þ ð 3 Þ  2 2  2ðn  5Þ; if n  4ðmod > > 3 3 > > n5 n2 :  2 þ ðn2 Þ2  ðn5 Þ2  2ðn  5Þ; if n  5ðmod 3 2 8 13 23 ðn  18n þ 84Þ; if n  0ðmod 6Þ; > 12 > > > 1 > ðn  7Þðn  11Þ; if n  1ðmod 6Þ; > 12 > > > < 1 ðn  8Þðn  10Þ; if n  2ðmod 6Þ; 12 ¼ 1 > ðn  9Þ2 ; if n  3ðmod 6Þ; > 12 > > > 1 > ðn  8Þðn  10Þ; if n  4ðmod 6Þ; > > 12 > :1 ðn  7Þðn  11Þ; if n  5ðmod 6Þ: 12

6Þ; 6Þ; 6Þ; 6Þ; 6Þ; 6Þ:

Hence W P ðU 3 Þ ¼ W P ðU 5 Þ if n ¼ 11, and W P ðU 3 Þ > W P ðU 5 Þ if n P 12. Combining these with Theorem 3.5 and Lemma 3.6, the assertion follows. h Remark 2. From Theorem 3.7, it can be seen that the corresponding extremal unicyclic graphs of the maximum Wiener polarity index are different from that of the maximum second Zagreb index (see [8,9]). This once again verifies: only use the relations between Zagreb indices and Wiener polarity index (see Remark 1) and the extremal unicyclic graphs of Zagreb indices is unable to deduce the results of this paper. Acknowledgements The authors are grateful to the referees for their helpful comments and useful suggestions on the earlier versions of this paper. The authors thank Dr. Muhuo Liu for providing many constructive comments. References [1] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17–20. [2] I. Lukovits, W. Linert, Polarity-numbers of cycle-containing structures, J. Chem. Inform. Comput. Sci. 38 (1998) 715–719. [3] H. Hosoya, Mathematical and chemical analysis of Wiener’s polarity number, in: D.H. Rouvray, R.B. King (Eds.), Topology in Chemistry–Discrete Mathematics of Molecules, vol. 57, Horwood, Chichester, 2002. [4] M. Liu, B. Liu, On the Wiener polarity index, MATCH Commun. Math. Comput. Chem. 66 (2011) 293–304. [5] H. Deng, On the extremal Wiener polarity index of chemical trees, MATCH Commun. Math. Comput. Chem. 66 (2011) 305–314. [6] W. Du, X. Li, Y. Shi, Algorithms and extremal problem on Wiener polarity index, MATCH Commun. Math. Comput. Chem. 62 (2009) 235–244. [7] I. Gutman, N. Trinajstic´, Graph theory and molecular orbitals, total p-electron energy of alternate hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538. [8] F. Xia, S. Chen, Ordering unicyclic graphs with respect to Zagreb indices, MATCH Commun. Math. Comput. Chem. 58 (2007) 663–673. [9] Z. Yan, H. Liu, H. Liu, Sharp bounds for the second Zagreb index of unicyclic graphs, J. Math. Chem. 42 (2010) 565–574.