Comparing the Zagreb indices of the NEPS of graphs

Applied Mathematics and Computation 219 (2012) 1082–1086

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Comparing the Zagreb indices of the NEPS of graphs q Dragan Stevanovic´ ⇑ University of Niš, Faculty of Sciences and Mathematics, Višegradska 33, 18000 Niš, Serbia FAMNIT—University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia

a r t i c l e

i n f o

a b s t r a c t P The first and the second Zagreb indices of a graph G ¼ ðV; EÞ are defined as M1 ðGÞ ¼ u2V P 2 dG ðuÞ and M 2 ðGÞ ¼ uv 2 EdG ðuÞdG ðv Þ, where dG ðuÞ denotes the degree of a vertex u in G. It has recently been conjectured that M 1 ðGÞ=jVj 6 M 2 ðGÞ=jEj. Although some counterexamples have already been found, the question of characterizing graphs for which the inequality holds is left open. We show that this inequality is preserved under the NEPS of graphs, while its opposite is preserved under the direct product of graphs. Ó 2012 Elsevier Inc. All rights reserved.

Keywords: The first Zagreb index The second Zagreb index Hansen–Vukicˇevic´ conjecture NEPS of graphs Cartesian product of graphs Direct product of graphs

1. Introduction Let G ¼ ðV; EÞ be a simple graph with n ¼ jVj vertices and m ¼ jEj edges. The first and the second Zagreb indices of G are P P defined as M 1 ðGÞ ¼ u2V dG ðuÞ2 and M 2 ðGÞ ¼ uv 2 EdG ðuÞdG ðv Þ, where dG ðuÞ denotes the degree of a vertex u in G. The Zagreb indices were introduced by Gutman and Trinajstic´ in [1], the surveys of their chemical importance and mathematical properties appear in [2,3], and some recent developments are given in [4–9].  ¼ 2m=n be the average degree of G. For general graphs of order n, the order of magnitude of M 1 is Hðnd 2 Þ, while the Let d 2 Þ, implying that M 1 =n and M 2 =m have the same order of magnitude. Furthermore, with order of magnitude of M2 is Hðmd P ð2Þ dG ðuÞ ¼ uv 2E dG ðv Þ denoting the 2-degree of vertex u, it was observed in [10] that

M1 ðGÞ X 1 ð2Þ ¼ d ðuÞ; n n G u2V

M2 ðGÞ X dG ðuÞ ð2Þ ¼ d ðuÞ: m 2m G u2V

P P G ðuÞ Since u2V 1n ¼ u2V d2m ¼ 1, this means that both expressions M 1 ðGÞ=n and M2 ðGÞ=m are the convex linear combinations of ð2Þ the same set of values fdG ðuÞ : u 2 Vg, which outlines their similarity and shows that it is useful to compare M 1 =n and M 2 =m. The use of AutoGraphiX system [11] led Hansen and Vukicˇevic´ to the following conjecture. Conjecture 1 ([12]). For all simple connected graphs G

M1 ðGÞ M 2 ðGÞ 6 ; n m

ð1Þ

and the bound is tight for complete graphs. q Supported by the research Grant 174033 of the Serbian Ministry of Science and Environmental Protection and the research programme P1-0285 of the Slovenian Agency for Research. ⇑ Address: FAMNIT—University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia. E-mail address: [email protected]

0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.07.014

D. Stevanovic´ / Applied Mathematics and Computation 219 (2012) 1082–1086

1083

This conjecture has generated a lot of research recently, perhaps due to the fact that it is correct for some well-known graph classes (chemical graphs [12], trees [13], unicyclic graphs [14], subdivided graphs [15]), while having counterexamples in other graph classes (bicyclic graphs [15,16], graphs with large stars attached [10]). A survey of the developments on this conjecture is given in [17]. Still, the question of characterizing the graphs for which the inequality (1) holds is left open. Here we make progress in that direction by proving that the inequality (1) is preserved under a very general graph composition, the so-called NEPS of graphs, while its opposite is preserved under a special case of the NEPS, the direct product of graphs. Although the preservation of the opposite inequality gives a way to construct new counterexamples from the existing ones, the main point here is that the set of graphs satisfying the inequality (1), being closed under the NEPS of graphs, has a rich algebraic structure. Many graph compositions are defined on the Cartesian product of vertex sets of graphs using only equality and adjacency among corresponding vertices of these graphs. Most widely known are the Cartesian product and the direct product of graphs, which are the special cases of the NEPS of graphs. The NEPS of graphs was defined for the first time in [18], and the following definition is taken from [19] p. 66 ], with a minor modification. Definition 1. Let B be a set of binary n-tuples, i.e. B # f0; 1gn n fð0; . . . ; 0Þg such that for every i ¼ 1; . . . ; n there exists b 2 B with bi ¼ 1. The non-complete extended p-sum (NEPS) of graphs G1 ; . . . ; Gn with basis B, denoted by NEPSðG1 ; . . . ; Gn ; BÞ, is the graph with the vertex set VðG1 Þ  . . .  VðGn Þ, in which two vertices ðu1 ; . . . ; un Þ and ðv 1 ; . . . ; v n Þ are adjacent if and only if there exists ðb1 ; . . . ; bn Þ 2 B such that ui is adjacent to v i in Gi whenever bi ¼ 1, and ui ¼ v i whenever bi ¼ 0. In such case, we will say that the adjacency of ðu1 ; . . . ; un Þ and ðv 1 ; . . . ; v n Þ is determined by ðb1 ; . . . ; bn Þ. The Cartesian product of n graphs is obtained for B ¼ fe1 ; . . . ; en g, where ei ; i ¼ 1; . . . ; n is the n-dimensional vector with 1 at the coordinate i and 0 elsewhere, while the direct product of graphs is obtained for B ¼ fð1; . . . ; 1Þg. 2. The Zagreb indices of the NEPS of graphs The degree of the vertex u ¼ ðu1 ; . . . ; un Þ in NEPSðG1 ; . . . ; Gn ; BÞ is, according to [19], equal to

dG ðuÞ ¼

n XY

b

dGii ðui Þ:

ð2Þ

b2B i¼1

In order to state the results, we need to define two auxiliary functions Z 0 ðG; jÞ and Z 1 ðG; jÞ for G ¼ ðV; EÞ and j 2 f0; 1; 2g as follows:

8 for j ¼ 0; > < jVj; X j Z 0 ðG; jÞ ¼ dG ðuÞ ¼ 2jEj; for j ¼ 1; > : u2V M 1 ðGÞ; for j ¼ 2 and

8 for j ¼ 0; > < 2jEj; dj=2e bj=2c Z 1 ðG; jÞ ¼ dG ðuÞdG ðv Þ ¼ M 1 ðGÞ; for j ¼ 1; > : u2V;v 2V;uv 2E 2M 2 ðGÞ; for j ¼ 2: X

(Here for j ¼ 1 we have Z 1 ðG; 1Þ ¼

P

uv 2E ðdG ðuÞ

þ dG ðv ÞÞ ¼

P

2 u2V dG ðuÞ

¼ M 1 ðGÞ.)

Theorem 2. Let G ¼ NEPSðG1 ; . . . ; Gn ; BÞ, where Gi ¼ ðV i ; Ei Þ for i ¼ 1; . . . ; n; n P 2. Then

M1 ðGÞ ¼

n X Y

Z 0 ðGi ; bi þ ci Þ:

ðb;cÞ2B2 i¼1

Proof. From (2) we have

X 2 M1 ðGÞ ¼ dG ðuÞ ¼ u2G

¼

X

X

n XY

ðu1 ;...;un Þ2V 1 ...V n

b2B i¼1

n X Y

ðu1 ;...;un Þ2V 1 ...V n ðb;cÞ2BB i¼1

¼

n X Y ðb;cÞ2B2 i¼1

Z 0 ðGi ; bi þ ci Þ:

b þci

dGii

!2 b dGii ðui Þ

ðui Þ ¼

X

X

n XY

ðu1 ;...;un Þ2V 1 ...V n

b2B i¼1

¼ X

n Y

ðb;cÞ2B2 ðu1 ;...;un Þ2V 1 ...V n i¼1



b þci

dGii

! b dGii ðui Þ

ðui Þ ¼

n XY

c2B i¼1

n X X Y ðb;cÞ2B2 i¼1 ui 2V i

! ci

dGi ðui Þ b þci

dGii

ðui Þ

D. Stevanovic´ / Applied Mathematics and Computation 219 (2012) 1082–1086

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Theorem 3. Let G ¼ NEPSðG1 ; . . . ; Gn ; BÞ, where Gi ¼ ðV i ; Ei Þ for i ¼ 1; . . . ; n; n P 2. Then

M2 ðGÞ ¼

1 2

n X Y

Z ai ðGi ; bi þ ci Þ:

ða;b;cÞ2B3 i¼1

Proof. From (2) we have

X

M 2 ðGÞ ¼

dG ðuÞdG ðv Þ ¼

fu;v g2EðGÞ

1X 2 a2B

X ðu1 ; . . . ; un Þ 2 V 1  . . .  V n

n XY b dGii ðui Þ

!

n XY

c2B i¼1

b2B i¼1

! c

dGii ðv i Þ

ðv 1 ; . . . ; v n Þ 2 V 1  . . .  V n ui ¼ v i for ai ¼ 0 ui v i 2 Ei for ai ¼ 1 n   X Y X X 1 b c dGii ðui Þ dGii ðv i Þ ¼ 2 a2B ðu1 ; . . . ; un Þ 2 V 1  . . .  V n ðb;cÞ2BB i¼1 ðv 1 ; . . . ; v n Þ 2 V 1  . . .  V n ui ¼ v i for ai ¼ 0

¼

1 2

X ða;b;cÞ2B3

ui v i 2 Ei for ai ¼ 1 X ðu1 ; . . . ; un Þ 2 V 1  . . .  V n

n   1 Y b c dGii ðui Þ dGii ðv i Þ ¼ 2 i¼1

n X Y

X ui 2 V i ; v i 2 V i

ða;b;cÞ2B3 i¼1

ðv 1 ; . . . ; v n Þ 2 V 1  . . .  V n

ui ¼ v i for ai ¼ 0

ui ¼ v i for ai ¼ 0

ui v i 2 Ei for ai ¼ 1

b

c

dGii ðui Þ dGii ðv i Þ

ui v i 2 Ei for ai ¼ 1 ¼

1 2

n X Y

Z ai ðGi ; bi þ ci Þ:



ða;b;cÞ2B3 i¼1

Example 1. Let G be the direct product of G1 ; . . . ; Gn . From Theorems 2 and 3 we get

M1 ðGÞ ¼

n n Y Y Z 0 ðGi ; 2Þ ¼ M 1 ðGi Þ; i¼1

M2 ðGÞ ¼

ð3Þ

i¼1

n n Y 1Y Z 1 ðGi ; 2Þ ¼ 2n1 M 2 ðGi Þ: 2 i¼1 i¼1

ð4Þ

Example 2. Let G ¼ ðV; EÞ be the Cartesian product of G1 ¼ ðV 1 ; E1 Þ; . . . ; Gn ¼ ðV n ; En Þ. From Theorem 2 we get

M1 ðGÞ ¼

n X n Y n n n n X n n X X Y X Y Z 0 ðGi ; ðek þ el Þi Þ ¼ Z 0 ðGk ; 2Þ Z 0 ðGi ; 0Þ þ Z 0 ðGk ; 1ÞZ 0 ðGl ; 1Þ Z 0 ðGi ; 0Þ k¼1 l¼1 i¼1

k¼1

i¼1;i–k

k¼1 l¼1;l–k

i¼1;i–k;l

n n n X n n n n X n Y X Y X X X M 1 ðGk Þ jEk jjEl j M1 ðGk Þ jV i j þ 4jEk jjEl j jV i j ¼ jVj ¼ þ 4jVj : jV jV j k k jjV l j k¼1 i¼1;i–k k¼1 l¼1;l–k i¼1;i–k;l k¼1 k¼1 l¼1;l–k

while from Theorem 3 we get

M2 ðGÞ ¼

n X n X n Y n 1X Z ðe Þ ðGi ; ðek þ el Þi Þ 2 m¼1 k¼1 l¼1 i¼1 m i

¼

n X n n n X n X n n Y Y 1X 1X Z 1 ðGm ; ðek þ ek Þm Þ Z 0 ðGi ; ðek þ ek Þi Þ þ Z 1 ðGm ; ðek þ el Þm Þ Z 0 ðGi ; ðek þ el Þi Þ 2 m¼1 k¼1 2 m¼1 k¼1 l¼1;l–k i¼1;i–m i¼1;i–m

¼

n n n n n Y Y X 1X 1X Z 1 ðGm ; 2Þ Z 0 ðGi ; 0Þ þ Z 1 ðGm ; 0ÞZ 0 ðGk ; 2Þ Z 0 ðGi ; 0Þ 2 m¼1 2 m¼1k¼1;k–m i¼1;i–m i¼1;i–m;k

þ

n n n Y X 1X Z 1 ðGm ; 1ÞZ 0 ðGl ; 1Þ Z 0 ðGi ; 0Þ 2 m¼1l¼1;l–m i¼1;i–m;l

D. Stevanovic´ / Applied Mathematics and Computation 219 (2012) 1082–1086

þ

1085

n n n n n n n Y Y X X X 1X 1X Z 1 ðGm ; 1ÞZ 0 ðGk ; 1Þ Z 0 ðGi ; 0Þ þ Z 1 ðGm ; 0ÞZ 0 ðGk ; 1ÞZ 0 ðGl ; 1Þ Z 0 ðGi ; 0Þ 2 m¼1k¼1;k–m 2 m¼1k¼1;k–ml¼1;l–m;k i¼1;i–m;k i¼1;i–m;k;l ! n n n n n n X X X X X X M2 ðGm Þ M1 ðGm ÞjEk j jEm jjEk jjEl j ¼ jVj þ3 þ4 : jV m j jV m jjV k j jV m jjV k jjV l j m¼1 m¼1k¼1;k–m m¼1k¼1;k–ml¼1;l–m;k

jEj These two formulas were obtained recently in [20] in a different way. Considering that jV ¼ j of symmetric functions they can be further transformed into

M1 ðGÞ ¼ jVj

n X M 1 ðGm Þ

jV m j

m¼1

M2 ðGÞ ¼ jVj

n X M 2 ðGm Þ

jV m j

m¼1

þ

Pn

jEk j k¼1 jV k j,

with simple properties

n X 4jEj2 jEm j2 ;  4jVj 2 jVj m¼1 jV m j

þ3

  n n n X X X M 1 ðGm Þ jEm j 4jEj3 jEm j3 jEm j2 þ þ 8jVj  12jEj : jEj  jVj 2 3 2 jV m j jV m j jVj m¼1 m¼1 jV m j m¼1 jV m j

3. The NEPS of graphs preserves the inequality (1) 1 ðGi Þ 2 ðGi Þ 6 MjE for each i ¼ 1; . . . ; n, Theorem 4. Let G ¼ ðV; EÞ ¼ NEPSðG1 ; . . . ; Gn ; BÞ, where Gi ¼ ðV i ; Ei Þ for i ¼ 1; . . . ; n; n P 2. If MjV ij ij M 1 ðGÞ M 2 ðGÞ then jVj 6 jEj .

Proof. Since V ¼ V 1  . . .  V n , it holds that jVj ¼

X 2jEj ¼ dG ðuÞ ¼

X

Qn

ðu1 ;...;un Þ2V 1 ...V n a2B i¼1

u2V

i¼1 jV i j.

n X XY a dGii ðui Þ ¼

We calculate jEj from (2):

X

n Y

a2B ðu1 ;...;un Þ2V 1 ...V n i¼1

a

dGii ðui Þ ¼

n X XY

a2B i¼1 ui 2V i

a

dGii ðui Þ ¼

n XY

Z 0 ðGi ; ai Þ:

a2B i¼1

Next, we observe that

Z ai ðGi ; bi þ ci ÞjV i j P Z 0 ðGi ; ai ÞZ 0 ðGi ; bi þ ci Þ for each ða; b; cÞ 2 B3

and i ¼ 1; . . . ; n;

since

Z ai ðGi ; bi þ ci ÞjV i j  Z 0 ðGi ; ai ÞZ 0 ðGi ; bi þ ci Þ ¼

2

P

P 2 2 ui 2V i dGi ðui Þ ui 2V i 1

8 0 > > > > > 0 > > > > > 0 > > > > <0

ðai ; bi ; ci Þ ¼ ð0; 0; 0Þ; ðai ; bi ; ci Þ ¼ ð0; 0; 1Þ; ðai ; bi ; ci Þ ¼ ð0; 1; 0Þ; ðai ; bi ; ci Þ ¼ ð0; 1; 1Þ; ðai ; bi ; ci Þ ¼ ð1; 0; 0Þ;

0 > > > > 2 > > > M 1 ðGi ÞjV i j  4jEi j > > > > > M 1 ðGi ÞjV i j  4jEi j2 > > : 2M2 ðGi ÞjV i j  2jEi jM 1 ðGi Þ P

ðai ; bi ; ci Þ ¼ ð1; 0; 1Þ; ðai ; bi ; ci Þ ¼ ð1; 1; 0Þ; ðai ; bi ; ci Þ ¼ ð1; 1; 1Þ;

2

where M 1 ðGi ÞjV i j  4jEj ¼  P 0 by the Cauchy–Schwarz inequality and ui 2V i dGi ðui Þ 2jV i jM 2 ðGi Þ  2jEi jM 1 ðGi Þ P 0 by assumption. 1 ðGÞ 2 ðGÞ The inequality MjVj 6 MjEj follows now from the equivalent inequality 2M 2 ðGÞjVj P 2jEjM 1 ðGÞ. From Theorems 2 and 3 we have

0

1 n n X Y Y 2M 2 ðGÞjVj ¼ @ Z ai ðGi ; bi þ ci ÞA jV i j ¼ ða;b;cÞ2B3 i¼1

¼

i¼1

n X Y Z ai ðGi ; bi þ ci ÞjV i j P ða;b;cÞ2B3 i¼1

Z 0 ðGi ; ai ÞZ 0 ðGi ; bi þ ci Þ

ða;b;cÞ2B3 i¼1

1 !0 n n XY X Y Z 0 ðGi ; ai Þ @ Z 0 ðGi ; bi þ ci ÞA ¼ 2jEjM 1 ðGÞ: a2B i¼1

n X Y



ðb;cÞ2B2 i¼1

The following corollary follows easily from the proof of Theorem 4. 1 ðGi Þ 2 ðGi Þ > MjE for each Corollary 5. Let G ¼ ðV; EÞ be the direct product of G1 ; . . . ; Gn , where Gi ¼ ðV i ; Ei Þ for i ¼ 1; . . . ; n; n P 2. If MjV ij ij M1 ðGÞ M 2 ðGÞ i ¼ 1; . . . ; n, then jV j > jEj .

Proof. Since B ¼ fð1; . . . ; 1Þg for the direct product of graphs, we have 2jEj ¼ Qn Q Q 2n1 M ðGi Þ 1 ðGÞ 1 ðGi Þ 2 ðGi Þ 2 ðGÞ i¼1 2 we have MjV ¼ ni¼1 MjV > ni¼1 MjE ¼ n1 Q ¼ MjEj . h n j j j i

i

2

i¼1

jEi j

Qn

i¼1 Z 0 ðGi ; 1Þ

¼ 2n

Qn

i¼1 jEi j,

and from Example 1

1086

D. Stevanovic´ / Applied Mathematics and Computation 219 (2012) 1082–1086

It is easy to see that no other NEPS of graphs may preserve the opposite of the inequality (1) for arbitrary G1 ; . . . ; Gn , as the difference 2M2 ðGÞjVj  2jEjM 1 ðGÞ will, in such case, involve both the negative terms 2M 2 ðGi ÞjV i j  2jEi jM 1 ðGi Þ and the nonnegative terms M 1 ðGi ÞjV i j  4jEi j2 , the sign of whose sum will then depend on the choice of G1 ; . . . ; Gn . 4. Conclusion From Theorem 4 and Corollary 5 we see that the set A6 of the graphs satisfying (1) is closed under arbitrary NEPS of graphs, while the set A> of the counterexamples to Conjecture 1 is closed under the direct product of graphs only. Although A> – ; (see [17] for an extensive list of counterexample constructions), the difference in the behavior of A6 and A> indicates the possibility that Conjecture 1 may be valid for the majority of graphs, perhaps even for almost all graphs. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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