* Author to whom correspondence should be addressed. (E-mail: zhoubo@scnu.edu.cn)

CROATICA CHEMICA ACTA CCACAA, ISSN 0011-1643, e-ISSN 1334-417X

Croat. Chem. Acta 84 (4) (2011) 493–497. CCA-3504

Original Scientific Article

On Zagreb Eccentricity Indices

Rundan Xing,a Bo Zhou,a,* and Nenad Trinajstićb

aDepartment of Mathematics, South China Normal University, Guangzhou 510631, China bThe Rugjer Bošković Institute, P. O. Box 180, HR-10 002 Zagreb, Croatia

RECEIVED NOVEMBER 20, 2010; REVISED JUNE 1, 2011; ACCEPTED JUNE 2, 2011

Abstract. Zagreb eccentricity indices were proposed analogously to Zagreb indices already known and used for almost forty years. For a connected graph, the first Zagreb eccentricity index is defined as the sum of the squares of the eccentricities of the vertices, and the second Zagreb eccentricity index is defined as the sum of the products of the eccentricities of pairs of adjacent vertices. We report mathematical prop-erties, especially lower and upper bounds of trees and general graphs in terms of graph invariants and the corresponding extremal graphs, Nordhaus-Gaddum-type results, and the ordering of trees with small and large Zagreb eccentricity indices. (doi: 10.5562/cca1801)

Keywords: Zagreb indices, Zagreb eccentricity indices, lower and upper bounds, graph invariants, trees, general graphs

INTRODUCTION

Let G be a connected graph with vertex set V(G) and edge set E(G). For a vertex ( ),u V G ( )Ge u or ue

denotes the eccentricity of u in G, which is the length of a path from u to a vertex v that is farthest from u, i.e.,

max{ ( , ) : ( )},ue d u v | G v V G where ( , | )d u v G de-

notes the distance between u and v in G.1 The Zagreb eccentricity indices are introduced in

an analogy with the Zagreb indices2–4 by replacing the vertex degrees with the vertex eccentricities. Thus, the first Zagreb eccentricity index of a graph G is defined as:

21 1

( )

( ) ,uu V G

ξ ξ G e

and the second Zagreb eccentricity index of G is defined as:

2 2( )

( ) .u vuv E G

ξ ξ G e e

Note that degrees are "local properties", while eccentric-ities are "global properties" of the vertices.

These two indices are recently introduced by Vukičević and Graovac.5 However, it should be pointed out that there were a number of eccentricity-based mo-lecular descriptors already proposed in the literature,6,7 such as the eccentric connectivity index8,9 (see Refe-rences 10–12 for recent results) and the eccentric adja-cency index.13 Vukičević and Graovac have shown in

their paper5 that 1 2( ) ( )

| ( ) | | ( ) |

ξ G ξ G

V G E G if G is a tree or a

unicyclic graph but is not true for bicyclic graphs. In the present paper, we give lower and upper

bounds for the first and the second Zagreb eccentricity indices of n-vertex trees with fixed diameter, and lower bounds for the first and the second Zagreb eccentricity indices of n-vertex trees with fixed matching number, and characterize the extremal cases, and determine the n-vertex trees with respectively the minimum, second-minimum and third-minimum, as well as the maximum, second-maximum and third-maximum indices 1ξ and

2ξ for 6.n We also give lower and upper bounds for

the first and the second Zagreb eccentricity indices of connected graphs in terms of graph invariants such as the number of vertices, the number of edges, the radius and the diameter, and give the Nordhaus-Gaddum-type results.14

PRELIMINARIES

For a connected graph G, the radius r(G) and the diame-ter D(G) are, respectively, the minimum and maximum eccentricities among the vertices of G.1 A connected graph is called a self-centered graph if all of its vertices have the same eccentricity. Evidently, a connected graph G is self-centered if and only if ( ) ( ).r G D G

494 R. Xing et al., On Zagreb Eccentricity Indices

Croat. Chem. Acta 84 (2011) 493.

Let Kn be the n-vertex complete graph. Let Sn and

nP be the n-vertex star and path, respectively. By direct

calculation, 1( )nξ K n and 2

( 1)( )

2n

n nξ K

for

2,n 1( ) 4 3,nξ S n and 2 ( ) 2( 1)nξ S n for 3,n

and if 2n is even, then:

1 1 /2 12 2 2

1/ 2 1 1

3 2

22

2/ 2

22 / 2 1 22 2

1 1 /2

( ) 2 2

1 1( 1) (2 1) 1 ( 1)

3 3 2 2

7 9 2,

12

( ) 2 ( 1)2

24

1 1( 2)( 1)(2 3)

3 3 2

n n n

ni n i i

n

ni n

n n n

i i i n

ξ P i i i

n nn n n n

n n n

nξ P i i

ni i i

nn n n

2

3 2

1

3( 1) 1 2

2 2 2 4

7 21 20,

12

n n n nn

n n n

and if 3n is odd, then:

2 2( 1)/ 21 1

2 2 21

( 1)/2 1 1

2

3 2

( 3)/ 22 22 2

2( 1)/ 2 1 1 ( 1)

1 1( ) 2 2

2 2

1 1 ( 1) ( 1) 1( 1) (2 1)

3 3 2 2 2

7 9 3,

12

( ) 2 ( 1) 2

nn n

ni n i i

nn n

ni n i i i n

n nξ P i i i

n n nn n n n

n n n

ξ P i i i i i

2

/ 2

3 2

1 1 ( 3) ( 1)( 2)( 1)(2 3) ( 2)

3 3 2 2( 1) (3 5)

2 2

7 21 17 3.

12

n

n nn n n n

n n

n n n

For a graph G and a subset E' of its edge set (E* of the edge set of its complement, respectively), G – E' (G + E*, respectively) denotes the graph formed from G by deleting (adding, respectively) edges from E' (E* respectively). For a graph G with ( ),u V G G – u de-notes the graph formed from G by deleting the vertex u (and its incident edges).

RESULTS FOR TREES

In this section, we give lower and upper bounds for the first and the second Zagreb eccentricity indices of n-vertex trees with fixed diameter, and lower bounds for the first and the second Zagreb eccentricity indices of n-vertex trees with fixed matching number. We also de-termine the n-vertex trees with, respectively, the mini-mum, second-minimum and third-minimum, as well as the maximum, second-maximum and third-maximum indices 1ξ and 2ξ for 6.n

Lemma 1. Let u be a vertex of a tree Q with at

least two vertices. For integer 1,a let G1 be the tree obtained from Q by attaching a star Sa+1 at its center v to ,u G2 the tree obtained from Q by attaching a + 1 pendant vertices to ,u see Figure 1. Then 1 2( )ξ G

1 1( )ξ G and 2 2 2 1( ) ( ).ξ G ξ G Let ( , )n dT be the set of n-vertex trees with di-

ameter ,d where 2 1.d n For 2 2,d n let

,Ln d be the set of n-vertex trees obtained from

1 0 1d dP v v v by attaching 1n d pendant vertices to / 2dv

and/or /2 .dv Let , 1L { }.n n nP Note that in

,L ,n d there is only one tree for even ,d and 1

2

n d

trees for odd .d Let ,an dU be the n-vertex tree obtained

by attaching a and 1n a d pendant vertices, respectively, to the two end vertices of the path

1dP for 1

1 ,2

n da

and let Un,d , :1a

n dU a

1.

2

n d

See Figures 2–4 for the trees in ,Ln d and

the tree ,an dU .

Proposition 1. Let ( , ),T n dT where 2 d

1.n Then:

Figure 1. The trees G1 and G2 in Lemma 1.

Figure 2. The structure of trees in Ln,d with odd d.

R. Xing et al., On Zagreb Eccentricity Indices 495

Croat. Chem. Acta 84 (2011) 493.

2 3 2

1 2 3 2

3( 2) 4 3 22 12 if is even

12( )3( 3) 4 9 40 27

if is odd,12

d n d d dd

ξ Td n d d d

d

3 2

2 3 2

3 ( 2) 4 9 10 if is even

12( )3( 1)( 3) 4 15 22 3

if is odd12

d d n d d dd

ξ Td d n d d d

d

with either equality if and only if ,L .n dT

Proposition 2. Let ( , ),T n dT where 2 d 1.n Then:

32

1 32

5 2 if even

12( )5 5

if is odd,12

d dd n d is

ξ Td d

d n d

3

2

2 32

5 8( ) if is even

12( )5 11 6

( ) if is odd12

d dd d n d

ξ Td d

d d n d

with either equality if and only if ,U .n dT

Now we consider the n-vertex trees with fixed matching number. A matching M of the graph G is a subset of ( )E G such that no two edges in M share a common vertex. A matching M of G is said to be maximum, if for any other matching 1M of ,G

1| | | | .M M The matching number of G is the number of edges of a maximum matching in .G If M is a matching of a graph G and vertex ( )v V G is inci-dent with an edge of M , then v is said to be M-saturated, and if every vertex of G is M-saturated, then M is a perfect matching. For integers n and 1 / 2 ,m n let ( , )n mT be the set of n-vertex trees

with matching number .m Obviously, ( ,1) { }.n nST For 3 m / 2 ,n let ,n mT be the tree obtained by attaching 1m paths on two vertices to the center of the star 2 2 ,n mS see Figure 5.

Proposition 3. Let (2 , )m mT T with 3.m Then

1( ) 25 12,ξ T m 2 ( ) 18 12ξ T m with either equality if and only if 2 , .m mT T

Proposition 4. Let ( , )n mT T with 2 ≤ m ≤ |n/2|.

If 2,m then 1( ) 9 10ξ T n and 2 ( ) 6 8ξ T n with either equality if and only if ( ,3)T nT ; If 3,m then 1( ) 9 7 12ξ T n m and 2 ( ) 6ξ T n 6 12m with either equality if and only if , .n mT T

Now we use the result for the first and the second Zagreb eccentricity indices of the n-vertex trees with fixed diameter to determine the n-vertex trees with small and large indices 1ξ and 2 .ξ

Proposition 5. Among the n-vertex trees, nS for

3,n the trees in ,3Ln for 4,n and the trees in

,4Ln for 5n are, respectively, the unique trees with the minimum, second-minimum and third-minimum indices 1ξ and 2 ,ξ the first Zagreb eccentricity indic-es of which are equal to 4 3,n 9 10n and 9 9,n respectively, while the second Zagreb eccentricity indices of which are equal to 2 2,n 6 8n and 6 6,n respectively.

For 4,n let nP be the tree formed by attaching a pendant vertex to the neighbor of an end-vertex of the path 1.nP For 6,n let nP be the tree formed by at-taching a pendant vertex to the second neighbor of an end-vertex of the path 1.nP

Proposition 6. Among the n-vertex trees, nP for

3,n nP for 4,n and nP for 5n are, respective-ly, the unique trees with the maximum, second-maximum and third-maximum indices 1ξ and 2 ,ξ where:

3 2

1 3 2

7 9 2 if is even

12( ) 7 9 3

if is odd,12

n

n n nn

ξ Pn n n

n

Figure 3. The tree in Ln,d with even d.

Figure 4. The tree , .an dU

Figure 5. The tree Tn,m.

496 R. Xing et al., On Zagreb Eccentricity Indices

Croat. Chem. Acta 84 (2011) 493.

3 2

2 3 2

3 2

1 3 2

3 2

2

7 21 20 if is even

12( )7 21 17 3

if is odd,12

7 18 10 36 if is even

12( ) 7 18 7 30

if is odd,12

7 30 20 24 if is even

12( )

n

n

n

n n nn

ξ Pn n n

n

n n nn

ξ Pn n n

n

n n nn

ξ P

3 2

3 2

1 3 2

3 2

2 3 2

7 30 23 24 if is odd,

12

7 18 34 96 if is even

12( ) 7 18 31 90

if is odd,12

7 30 4 96 if is even

12( )7 30 96

if is odd.12

n

n

n n nn

n n nn

ξ Pn n n

n

n n nn

ξ Pn n n

n

RESULTS FOR GENERAL CONNECTED GRAPHS

In this section, we give lower and upper bounds for the first and the second Zagreb eccentricity indices of con-nected graphs in terms of graph invariants such as the number of vertices, the number of edges, the radius and the diameter, and we also come up with the Nordhaus-Gaddum-type results.14 Moreover, among the n-vertex connected graphs, we establish lower and upper bounds for the first Zagreb eccentricity index, and lower bound for the second Zagreb eccentricity index, respectively, and characterize the extremal cases.

Proposition 7. Let G be a connected graph with

n vertices and m edges. Then:

2 2

1

2 22

( ) ( ) ( ) ,

( ) ( ) ( )

nr G ξ G nD G

mr G ξ G mD G

with any equality if and only if G is a self-centered graph. Proposition 8. Let G be a connected graph with

2n vertices. Then:

1 1 1( ) ( ) ( )n nξ K ξ G ξ P

with left equality if and only if nG K and right equali-ty if and only if .nG P

Corollary 1. Let G be a connected bipartite graph

with 3n vertices. Then 1( ) 4 3ξ G n with equality if and only if .nG S

Proposition 9. Let G be a connected graph with

2n vertices. Then:

2

1 2

( ) 3 3

2( 1) 4

if n

ξ G if n

n if n

with equality if and only if nG K for 2,3,n

4G K or 4S for 4,n and nG S for 5.n Remark. By Proposition 8, among the n-vertex

connected graphs, nP is the unique graph with the max-imum first Zagreb eccentricity index. However, nP is not the graph with the maximum second Zagreb eccen-tricity index for 9n . For 5,n let nB be the bicyclic graph obtained by attaching a path 4nP (at one end vertex) to a vertex of degree two in the bicyclic graph obtained by adding an edge to the quadrangle, see Fig-ure 6. Then:

2

2 2 1

3 2

3 2

( ) ( ) ( 2)( 3) ( 3) ( 3)( 4)

7 6 136 276 if is even

12 7 6 133 276

if is odd.12

n nξ B ξ P n n n n n

n n nn

n n nn

For 9n :

2 2

3 2 3 2

3 2 3 2

2

2

( ) ( )

7 6 136 276 7 21 20 if is even

12 12

7 6 133 276 7 21 17 3 if is odd

12 12

15 156 276 if is even

12

15 150 279 if is odd

120,

n nξ B ξ P

n n n n n nn

n n n n n nn

n nn

n nn

and then 2 2( ) ( ).n nξ B ξ P

Figure 6. The graph Bn.

R. Xing et al., On Zagreb Eccentricity Indices 497

Croat. Chem. Acta 84 (2011) 493.

Denote by G the complement of a graph .G There is only one connected graph 4P on four vertices with the connected complement 4 4 .P P Obviously, we have 1 4 1 4 1 4( ) ( ) 2 ( ) 52.ξ P ξ P ξ P For 5,n the diameter of nP is two.

Now we give the Nordhaus-Gaddum-type results. Proposition 10. Let G be a connected graph with

4n vertices for which G is also connected. Then: 1 1 1 18 ( ) ( ) ( ) ( )n nn ξ G ξ G ξ P ξ P

with left equality if and only if G and G are self-centered graphs with radius two and right equality if and only if nG P or ,nP and 2 2( ) ( ) 2 ( 1)ξ G ξ G n n with equality if and only if G and G are self-centered graphs with radius two.

CONCLUSION

In this paper, we study the novel molecular descriptors, called Zagreb eccentricity indices, which were proposed analogously to Zagreb indices already known and used for almost forty years. We give lower and upper bounds for Zagreb eccentricity indices of trees and general graphs in terms of graph invariants, and characterize the extrem-al graphs, and determine the n-vertex trees with small and large Zagreb eccentricity indices. We also give Nord-haus-Gaddum-type results for Zagreb eccentricity indic-es. The lower and upper bounds for a molecular descrip-tor in terms of some graph invariants (e.g., the number of vertices) are important information for a molecule in the sense that they establish the range of the descriptor.

Supplementary Materials. – Supporting informations to the paper are enclosed to the electronic version of the article.

These data can be found on the website of Croatica Chemica Acta (http://public.carnet.hr/ccacaa).

Acknowledgements. This work is supported by the National Natural Science Foundation of China (Grant No. 11071089) and the Ministry of Science, Education and Sports of Croatia (Grant No. 098-1770945-2919).

REFERENCES

1. F. Harary, Graph Theory, 2nd printing, Addison-Wesley, Read-ing, MA, 1971, p. 35.

2. I. Gutman and N. Trinajstić, Chem. Phys. Lett. 17 (1972) 535–538.

3. I. Gutman, B. Ruščić, N. Trinajstić, and C.F. Wilcox, Jr., J. Chem. Phys. 62 (1975) 3399–3405.

4. S. Nikolić, G. Kovačević, A. Miličević, and N. Trinajstić, Croat. Chem. Acta 76 (2003) 113–124.

5. D. Vukičević and A. Graovac, Acta Chim. Slov. 57 (2010) 524–538. 6. R. Todeschini and V. Consonni, Handbook of Molecular De-

scriptors, Wiley-VCH, Weinheim, 2000, p. 124. 7. R. Todeschini and V. Consonni, Molecular Descriptors for

Chemoinformatics, Vol. I, Wiley-VCH, Weinheim, 2009, pp. 237–241.

8. V. Sharma, R. Goswami, and A. K. Madan, J. Chem. Inf. Com-put. Sci. 37 (1997) 273–282.

9. S. Sardana and A. K. Madan, J. Comput. Aid. Mol. Des. 16 (2002) 545–550.

10. B. Zhou and Z. Du, MATCH Commun. Math. Comput. Chem. 63 (2010) 181–198.

11. A. Ilić, In: I. Gutman and B. Furtula (Eds.), Novel Molecular Structure Descriptors - Theory and Applications II, Univ. Kragu-jevac, Kragujevac, 2010, pp. 139–168.

12. M. Saheli and A. R. Ashrafi, J. Comput. Theor. Nanosci. 7 (2010) 1900–1903.

13. S. Gupta, M. Singh, and A. K. Madan, J. Comput. Aid. Mol. Des. 15 (2001) 671–678.

14. E. A. Nordhaus and J. W. Gaddum, Amer. Math. Mon. 63 (1956) 175–177.

15. R. A. Brualdi and J. L. Goldwasser, Discr. Math. 48 (1984) 1–21.

On Zagreb Eccentricity Indices

RUNDAN XING,A BO ZHOU,A,* AND NENAD TRINAJSTIĆB

SUPPLEMENTARY MATERIALS

Proof of Lemma 1

Denote by w a pendant neighbor of v in 1G and a pendant neighbor of u in 2G outside .Q It is easily seen that

2 1( ) ( )G Ge x e x for ( ),x V Q

2 1( ) ( )G Ge w e v and

1 1 1( ) ( ) ( ).G G Ge u e v e w Then:

2 1 2 1 1

1 1 1

1 1

2 2 2 2 21 2 1 1

( )

2 2 2

2 2

( ) ( ) [ ( ) ( ) ] ( 1) ( ) ( ) ( )

( 1) ( ) ( ) ( )

[ ( ) ( ) ] 0,

G G G G Gx V Q

G G G

G G

ξ G ξ G e x e x a e w ae w e v

a e v ae w e v

a e v e w

2 2 1 1

2 2 1 1 1 1

1 2 1 1 2 1

1 1 1

2 2 2 1( )

( ) ( ) [ ( ) ( ) ( ) ( )]

( 1) ( ) ( ) ( ) ( ) ( ) ( )

( )[ ( ) ( )] ( )[ ( ) ( )]

( )[ ( ) ( )] 0,

G G G Gxy E Q

G G G G G G

G G G G G G

G G G

ξ G ξ G e x e y e x e y

a e u e w a e v e w e u e v

ae v e u e w e v e u e u

ae v e u e w

and thus 1 2 1 1( ) ( )ξ G ξ G and 2 2 2 1( ) ( ).ξ G ξ G

Proof of Proposition 1

The cases 2, 1d n are trivial. Suppose that 3 2d n and T is a tree in ( , )n dT with the minimum indices 1ξ or 2 .ξ Let

0 1( ) dP T v v v be a diametrical path of .T By Lemma 1, all vertices outside ( )P T are pendant vertices adjacent to vertices of ( ).P T Suppose that there exists some vertex kv with / 2 , / 2 ,k d d such that the degree of kv is greater than two. Suppose without loss of generality that / 2 1,d k d i.e., ( ) .G ke v k Denote by 1,..., tw w the pendant neighbors of kv outside ( ).P T For 1 1 1 1 1{ ,..., } { ,..., } ( , ),k k t k k tT T v w v w v w v w n d T we have:

2 2

1 1 1( ) ( ) ( 1) (2 1) 0,ξ T ξ T t k t k k t

2 1 2( ) ( ) ( 1) ( 1) 2 0,ξ T ξ T t k k t k k kt

and thus 1 1 1( ) ( )ξ T ξ T and 2 1 2( ) ( ),ξ T ξ T a contradiction. It follows that the degree of iv is equal to two for all 1 1i d with / 2 , / 2 .i d d Thus ,L .n dT

Note that for ,L ,n dT 21 1 1( ) ( ) ( 1) ( / 2 1)dξ T ξ P n d d and 2 2 1( ) ( ) ( 1)dξ T ξ P n d

/ 2 ( / 2 1).d d By the expressions for 1( )i dξ P with 1, 2i (in Section 2) we may easily have the expressions for 1( )ξ T and 2 ( )ξ T in terms of n and d for ,L .n dT The result follows.

Proof of Proposition 2

The cases 2, 1d n are trivial. Suppose that 3 2d n and T is a tree in ( , )n dT with the maximum indices 1ξ or 2ξ . Let

0 1( ) dP T v v v be a diametrical path of .T Suppose that ,U ,n dT i.e., there exists some vertex kv with 1, 1,k d whose degree is greater than two. Suppose without loss of generality that / 2 1,d k d i.e.,

( ) .G ke v k Let kT be the branch of 1 1{ , }k k k kT v v v v containing .kv Let y be a pendant vertex of kT different

from ,kv and denote by x the neighbor of .y Then ( ) ,Te y d while ( ) ( ) 1 1.T Te x e y d For

1 1{ } { } ( , ),T T xy v y n d T we have:

1

2 2 2 21 1 1( ) ( ) ( ) ( ) ( ) 0,T T Tξ T ξ T e y e y d e y

1 12 1 2 1( ) ( ) ( ) ( ) ( ) ( ) ( 1) ( )[ ( ) 1] 0T T T T T Tξ T ξ T e v e y e x e y d d e y e y

with either equality if and only if ( ) ( ) 1 .T Te y e x d Continuing this procedure, we may finally obtain a tree in ( , )n dT with larger indices 1ξ and 2 ,ξ which is a contradiction. Thus ,U .n dT

It is easily seen that for ,U ,n dT 21 1 1( ) ( ) ( 1)dξ T ξ P n d d and 2 2 1( ) ( ) ( 1) ( 1).dξ T ξ P n d d d

Now by the expressions for 1( )i dξ P with 1, 2i (in Section 2), we get the desired bounds.

Proof of Proposition 3

We prove the result by induction on .m The case 3m is easy to check. Now suppose that 4m and the result holds for trees in (2( 1), 1) .m m T Let (2 , )m mT T with a perfect matching

.M Let u be a pendant vertex of a diametrical path of .T Obviously, the unique neighbor v of u has degree two. Then uv M and (2( 1), 1) .m mT u v T Let w be the neighbor of v different from .u Note that ( ) 2.Te w Then

( ) ( ) 1 ( ) 2 4.T T Te u e v e w By the induction hypothesis, we have:

2 21 1

2 21

( ) ( ) ( ) ( )

( ) 4 3

25( 1) 12 25

25 12,

T Tξ T ξ T u v e u e v

ξ T u v

m

m

2 2

2

( ) ( ) ( ) ( ) ( ) ( )

( ) 4 3 3 2

18( 1) 12 18

18 12

T T T Tξ T ξ T u v e u e v e v e w

ξ T u v

m

m

with either equality if and only if ( ) ( )T u v Te x e x for all ( ) \{ , },x V T u v ( ) 4,Te u ( ) 3Te v and ( ) 2,Te w and 2( 1), 1 ,m mT u v T i.e., 2 , .m mT T The result follows.

Proof of Proposition 4

We prove the result by induction on n (for fixed m ). If 2 ,n m then the result follows from direct check for m = 2 and Proposition 3 for m ≥ 3.

Suppose that 2n m and the result holds for trees in ( 1, ) .n mT Let ( , ) .n mT T Then there is a maximum matching M and a pendant vertex u such that u is not M-saturated.15 Then ( 1, ) .n mT u T By the induction hypothesis, 1( ) 9( 1) 10ξ T u n and 2 ( ) 6( 1) 8ξ T u n with either equality if and only if ( 1,3)T u n T for 2,m and 1( ) 9( 1) 7 12ξ T u n m and 2 ( ) 6( 1) 6 12ξ T u n m with either equality if and only if

1,n mT T for 3.m Let v be the unique neighbor of .u Note that the radius of T is at least two. We have ( ) ( ) 1 3.T Te u e v If 2,m then:

21 1

21

( ) ( ) ( )

( ) 3

9( 1) 10 9

9 10,

Tξ T ξ T u e u

ξ T u

n

n

2 2

2

( ) ( ) ( ) ( )

( ) 3 2

6( 1) 8 6

6 8

T Tξ T ξ T u e u e v

ξ T u

n

n

with either equality if and only if ( ) ( )T u Te x e x for all ( ) \{ },x V T u ( ) 3,Te u ( ) 2Te v and

( 1,3),T u n T i.e., ( ,3).T nT If 3,m then:

21 1

21

( ) ( ) ( )

( ) 3

9( 1) 7 12 9

9 7 12,

Tξ T ξ T u e u

ξ T u

n m

n m

2 2

2

( ) ( ) ( ) ( )

( ) 3 2

6( 1) 6 12 6

6 6 12

T Tξ T ξ T u e u e v

ξ T u

n m

n m

with either equality if and only if ( ) ( )T u Te x e x for all ( ) \{ },x V T u ( ) 3,Te u ( ) 2Te v and 1, ,n mT u T i.e., , .n mT T The result follows.

Proof of Proposition 5

By Proposition 1, among the trees in ( , )n dT with 2 1,d n the trees in ,Ln d are the unique trees with the minimum indices 1ξ and 2 .ξ

Let 1 1( ) ( )f d ξ G and 2 2( ) ( ),f d ξ G where ,L .n dG Note that 1.n d If d is even, then:

2 3 2

1 1

2 3 2

2

2

2

3( 4) 4( 1) 9( 1) 40( 1) 27( 1) ( )

12

3( 2) 4 3 22 12

12

4 10( 3)

2

4 10( 3)( 1)

2

3 4 40,

2

d n d d df d f d

d n d d d

d dd n

d dd d

d d

3 2

2 2

3 2

2

2

3( 2)( 4) 4( 1) 15( 1) 22( 1) 3( 1) ( )

12

3 ( 2) 4 9 10

12

2( 2) 5 6

2

3 20,

2

d d n d d df d f d

d d n d d d

d n d d

d d

and if d is odd, then:

2 3 2

1 1

2 3 2

2

3( 3) 4( 1) 3( 1) 22( 1) 12( 1) ( )

12

3( 3) 4 9 40 27

12

9 12 30,

6

d n d d df d f d

d n d d d

d d

3 2

2 2

3 2

2

3( 1)( 3) 4( 1) 9( 1) 10( 1)( 1) ( )

12

3( 1)( 3) 4 15 22 3

12

3 20.

2

d d n d d df d f d

d d n d d d

d d

Thus, for any ( , )T n dT with 5,d we have ( ) ( ) (5) (4) (3) (2)i i i i i iξ T f d f f f f for 1, 2.i Note

that ( , 2) { }nn ST and ,3( ,3) L .nn T It follows that among the n-vertex trees, nS for 3,n the trees in ,3Ln for 4,n and the trees in ,4Ln for 5n are, respectively, the unique trees with the minimum, second-minimum and

third-minimum indices 1ξ and 2 .ξ It is easily seen that for ,nS the trees in ,3Ln and the trees in ,4L ,n their first Zagreb eccentricity indices are equal to 4 3,n 9 10n and 9 9,n respectively, and their second Zagreb eccentricity indices are equal to 2 2,n 6 8n and 6 6,n respectively.

Proof of Proposition 6

By Proposition 2, among the trees in ( , )n dT with 2 1,d n the tree(s) in ,Un d is (are) the unique tree(s) with the maximum indices 1ξ and 2 .ξ

Let 1 1( ) ( )g d ξ G and 2 2( ) ( ),g d ξ G where ,U .n dG Note that 1.n d If d is even, then:

3 32 2

1 1

2

2

2

5( 1) 5( 1) 5 2( 1) ( ) ( 1)

12 12

5 4(2 1)

4

5 4(2 1)( 1)

4

3 8 40,

4

d d d dg d g d d n d n

d dd n

d dd d

d d

3 3

2 22 2

2

2

5( 1) 11( 1) 6 5 8( 1) ( ) [( 1) ( 1)] ( )

12 12

5 4 42

4

3 4 40,

4

d d d dg d g d d d n d d n

d ddn

d d

and if d is odd, then:

3 32 2

1 1

2

2

2

5( 1) 2( 1) 5 5( 1) ( ) ( 1)

12 12

5 6 1(2 1)

4

5 6 1(2 1)( 1)

4

3( 1)0,

4

d d d dg d g d d n d n

d dd n

d dd d

d

3 3

2 22 2

2

2

5( 1) 8( 1) 5 11 6( 1) ( ) [( 1) ( 1)] ( )

12 12

5 6 12

4

3 2 10.

4

d d d dg d g d d d n d d n

d ddn

d d

Thus, for any ( , )T n dT with 3,d n we have ( ) ( ) ( 3) ( 2) ( 1)i i i i iξ T g d g n g n g n for 1, 2.i

Note that , 1U { }n n nP and , 2U { }.n n nP Then nP for 3n and nP for 4n are, respectively, the unique trees with the maximum and second-maximum indices 1ξ and 2 .ξ The expressions for 1( )nξ P and 2 ( )nξ P are given in Section 2. Note that 2

1 1 1( ) ( ) ( 2) ,n nξ P ξ P n 2 2 1( ) ( ) ( 2)( 3).n nξ P ξ P n n The expressions for 1( )nξ P and

2 ( )nξ P follow. The third-maximum 1ξ and 2ξ indices are achieved by the maximum value of ( )iξ T where

, 2( , 2) \ Un nT n n T and ( )iξ T where , 3Un nT for 1, 2.i By the transformation used in the proof of Proposition 2, we have ( ) ( )i i nξ T ξ P for , 2( , 2) \ Un nT n n T for 1, 2i with equality if and only if .nT P By direct calculation, we obtain:

3 2

21 1 1 3 2

7 18 34 96

12( ) ( ) ( 3)7 18 31 90

,12

n n

n n nif n is even

ξ P ξ P nn n n

if n is odd

3 2

2 2 1 3 2

7 30 4 96

12( ) ( ) ( 3)( 4)7 30 96

.12

n n

n n nif n is even

ξ P ξ P n nn n n

if n is odd

By Proposition 2, for , 3U ,n nT we have:

3

2

1 32

3 2

3 2

5( 3) 5( 3)( 3)

12( )5( 3) 2( 3)

( 3) 12

7 27 22 120

12

7 27 25 129 ,

12

n nn n if n is even

ξ Tn n

n n if n is odd

n n nif n is even

n n nif n is odd

32

2 32

3 2

3 2

5( 3) 11( 3) 6[( 3) ( 3)]

12( )5( 3) 8( 3)

[( 3) ( 3)] 12

7 39 20 108

12

7 39 17 111 .

12

n nn n n if n is even

ξ Tn n

n n n if n is odd

n n nif n is even

n n nif n is odd

Then for , 3Un nT with 6,n by direct comparison, we obtain:

3 2 3 2

1 1 3 2 3 2

2

2

7 18 34 96 7 27 22 120

12 12( ) ( )7 18 31 90 7 27 25 129

12 12

3 4 8

4

3 2 13

40,

n

n n n n n nif n is even

ξ P ξ Tn n n n n n

if n is odd

n nif n is even

n nif n is odd

3 2 3 2

2 2 3 2 3 2

2

2

7 30 4 96 7 39 20 108

12 12( ) ( )7 30 96 7 39 17 111

12 12

3 8 4

4

3 6 5

40.

n

n n n n n nif n is even

ξ P ξ Tn n n n n n

if n is odd

n nif n is even

n nif n is odd

Thus nP is the unique n-vertex tree with the third-maximum indices 1ξ and 2 .ξ The result follows.

Proof of Proposition 7

It is easily seen that 22 2

1( ) ( )

( ) ( ) ( )uu V G u V G

ξ G e D G nD G

with equality if and only if ( )ue D G for any

( ),u V G i.e., G is a self-centered graph. It is also easily seen that 2 2

2( ) ( )

( ) ( ) ( )u vuv E G uv E G

ξ G e e D G mD G

with equality if and only if ( )u ve e D G for any ( ),uv E G which is possible only when G is a self-centered

graph. Since ( )ue r G for ( ),u V G the lower bounds follow similarly.

Proof of Proposition 8

The case 2n is trivial. Now suppose that 3.n On one hand, it is easily seen that 2 21

( ) ( )

( ) 1uu V G u V G

ξ G e n

with equality if and only if every vertex in G has eccentricity one, i.e., .nG K On the other hand, it is easily seen

that for any ( ),e E G if { }G e is connected, then 1 1( ) ( { }).ξ G ξ G e Thus 1 1( ) ( )ξ G ξ T for a spanning tree T

of .G By Proposition 6, nP is the unique tree with the maximum 1.ξ Thus 1 1( ) ( )nξ G ξ P with equality if and only

if .nG P

Proof of Corollary 1

Let X and Y be the two-partite sets of G with | | | | .X Y If | | 2,X then ( ) 2Ge v for all ( ),v V G and thus

1 1( ) 4 4 3 ( ).nξ G n n ξ S The result follows.

Proof of Proposition 9

The cases 2,3n are trivial. Suppose that 4.n Let a be the number of vertices of eccentricity one. If ( ) 1,r G then 1 ,a n and

2

22

( 1) 3 1( ) 2 ( ) 2 min 2 2, 2 2

2 2 2 2 2

a a n nξ G a n a a n a n n

with equalities if and only if either 1a and 1n vertices are of eccentricity two and form an independent set, i.e.,

,nG S or 4n a for which 4 .G K If ( ) 2,r G then by Proposition 7, 2 ( ) 4 4( 1) 2( 1).ξ G m n n Now the result follows easily.

Proof of Proposition 10

Let m and m be respectively the number of edges of G and .G Note that both G and G have radius at least two, and then by Proposition 7:

1 1( ) ( ) 4 4 8 ,ξ G ξ G n n n 2 2( ) ( ) 4 4 2 ( 1)ξ G ξ G m m n n

with either equality if and only if G and G are self-centered, and ( ) ( ) 2.r G r G Then the lower bounds for ( ) ( )i iξ G ξ G with 1, 2i follow.

Note that either both G and G have diameter three, or one of them has diameter two. If ( ) ( ) 3,D G D G then by Proposition 7, 1 1( ) ( ) 9 9 18ξ G ξ G n n n with equality if and only if G and G are self-centered, and

( ) ( ) 3.D G D G If one of ( )D G and ( )D G is equal to two, say ( ) 2,D G then we have ( ) 2G

e v for all ( ),v V G and by Proposition 8, we have:

1 1 1 1 1( ) ( ) ( ) 4 ( ) ( )n n nξ G ξ G ξ P n ξ P ξ P

with equality if and only if .nG P If 5,n then the result follows easily by comparing the three possible values of

1 1( ) ( )ξ G ξ G where both G and G have diameter 3 with 1 5 1 5( ) ( ).ξ P ξ P If 6,n then it is easily checked that

1 1( ) ( ) 18 ,n nξ P ξ P n which implies the result.