Kyungpook Mathematical Journal 2018; 58(2): 221-229  
Bounds for the First Zagreb Eccentricity Index and First Zagreb Degree Eccentricity Index
P. Padmapriya* and Veena Mathad
Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore - 570 006, India, e-mail: padmapriyap7@gmail.com and veena_mathad@rediffmail.com
*Corresponding Author.
Received: August 28, 2017; Accepted: June 8, 2018; Published online: June 23, 2018.
© Kyungpook Mathematical Journal. All rights reserved.

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Abstract

The first Zagreb eccentricity index E1(G) of a graph G is defined as the sum of squares of the eccentricities of the vertices. In this paper some bounds for the first Zagreb eccentricity index and first Zagreb degree eccentricity index are computed.

Keywords: eccentricity, diameter, radius, Zagreb eccentricity indices, total eccentricity of a graph
1. Introduction

A systematic study of topological indices is one of the most striking aspects in many branches of mathematics with its applications and various other fields of science and technology. A topological index is a numeric quantity from the structural graph of a molecule. According to the IUPAC definition,[15] a topological index (or molecular structure descriptor) is a numerical value associated with chemical constitution for correlation of chemical structure with various physical properties, chemical reactivity or biological activity.

All the graphs G = (V,E) considered in this paper are simple, undirected and connected graphs. The number of vertices of G is denoted by n and the number of edges is denoted by m, thus |V (G)| = n and |E(G)| = m. For any vertices u, vV (G), the distance d(u, v) is defined as the length of any shortest path connecting u and v in G. For any vertex vi in G, the degree (di or d(vi)) of vi is the number of edges incident with vi in G. Especially, Δ = Δ(G) and δ = δ(G) are called the maximum and minimum degree of G, respectively. The eccentricity (ei or e(vi)) of vi is the largest distance between vi and any other vertex of G. The radius r = r(G) is the minimum eccentricity among the vertices of G. The diameter d = d(G) is the maximum eccentricity among the vertices of G. Also the second maximum eccentricity is written as d2. A graph G is equi-eccentric if the eccentricity of every vertex is same [9].

The Zagreb indices were introduced by Gutman and Trinajstić in 1972 [8]. The main properties of M1(G) and M2(G) were summarized in [12]. The first Zagreb index M1(G) of G is defined as M1(G)=viV(G)di2. The second Zagreb index M2(G) of G is defined as M2(G)=vivjE(G)didj.

The invariants based on vertex eccentricities attracted some attention in Chemistry. In an analogy with the first and the second Zagreb indices, M. Ghorbani et al. and D. Vukičević et al. introduced the Zagreb eccentricity indices [7, 14]. The first Zagreb eccentricity index (E1) of a graph G is defined as E1(G)=viV(G)ei2.

The Zagreb degree eccentricity indices are introduced in [13]. First Zagreb degree eccentricity index (DE1) of a graph G is defined as DE1=viV(G)(ei+di)2.

The total eccentricity index of G is defined as ζ(G)=viV(G)ei. Fathalikhani et al. [6] have studied total eccentricity of some graph operations. Nilanjan De et al. [4] have studied total eccentricity index of the Generalized Hierarchical Product of Graphs.

In this paper we obtain some bounds for the first Zagreb eccentricity index and first Zagreb degree eccentricity index.

2. Main Results

Theorem 2.1

([1]) Suppose ai and bi 1 ≤ in are positive real numbers, then

|ni=1naibi-i=1naii=1nbi|α(n)(A-b)(B-b)

where a, b, A and B are real constants, that for each i, 1 ≤ in, aaiA and bbiB. Further, α(n)=nn2(1-1nn2), wherexlargest integer greater than or equal to x.

We can see the appearance of Theorem 2.1, in [10].

Theorem 2.2

Let G be a nontrivial graph of order n, then

E1(G)α(n)(d-r)2+[ζ(G)]2n.

Further, equality holds if and only if G is a equi-eccentric graph.

Proof

Let a1, a2, …, an and b1, b2, …, bn be real numbers for which there exist real constants a, b, A and B, so that for each i, i = 1, 2, …, n, aaiA and bbiB. Then by Theorem 2.1, the following inequality is valid

|ni=1naibi-i=1naii=1nbi|α(n)(A-b)(B-b)

where α(n)=nn2(1-1nn2). Equality holds if and only if a1 = a2 = ⋯ = an and b1 = b2 = ⋯ = bn.

We choose ai = ei = bi, A = d = B and a = r = b, inequality (2.2) becomes

ni=1nei2-(i=1nei)2α(n)(d-r)(d-r)nE1(G)α(n)(d-r)2+[ζ(G)]2E1(G)α(n)(d-r)2+[ζ(G)]2n.

Since equality in (2.2) holds if and only if a1 = a2 = ⋯ = an and b1 = b2 = ⋯ = bn, it follows that for each vertex of a graph G has same eccentricity, equality of the theorem holds if and only if equi-eccentric graph.

Corollary 2.3

E1(G)n2(d-r)2+4[ζ(G)]24n.

Proof

Since α(n)n24, the proof follows by above theorem.

Theorem 2.4

Let G be a nontrivial graph of order n, then

E1(G)(r+d)ζ(G)-rdn.

Equality holds if and only if G is equi-eccentric graph.

Proof

Let a1, a2, …, an and b1, b2, …, bn be real numbers for which there exist real constants t and T, such that, taibiTai for each i, 1 ≤ in. Then the following inequality is valid (see [5])

i=1nbi2+tTi=1nai2(t+T)i=1naibi

Equality of (2.3) holds if and only if tai = bi = Tai for at least one i, 1 ≤ in

We choose bi = ei, ai = 1, t = r and T = d in inequality (2.3) then

i=1nei2+rdi=1n1(r+d)i=1neiE1(G)+rdn(r+d)ζ(G)E1(G)(r+d)ζ(G)-rdn.

If tai = bi = Tai for some i, then bi = t = T. Therefore equality holds if and only if r = d, that is G is a equi-eccentric graph.

Theorem 2.5

Let G be a nontrivial graph of order n and size m, then

DE1(G)α(n)(d+Δ-r-δ)2+[ζ(G)+2m]2n.
Proof

Let a1, a2, …, an and b1, b2, …, bn be real numbers for which there exist real constants a, b, A and B, so that for each i, i = 1, 2, …, n, aaiA and bbiB. Then by Theorem 2.1, the following inequality is valid

|ni=1naibi-i=1naii=1nbi|α(n)(A-b)(B-b)

Equality holds if and only if a1 = a2 = ⋯ = an and b1 = b2 = ⋯ = bn. We choose ai = ei + di = bi, A = d+Δ = B and a = r + δ = b, inequality (2.4) becomes

ni=1n(ei+di)2-[i=1n(ei+di)]2α(n)(d+Δ-r-δ)(d+Δ-r-δ)nDE1(G)α(n)(d+Δ-r-δ)2+[i=1nei+i=1ndi]2nDE1(G)α(n)(d+Δ-r-δ)2+[ζ(G)+2m]2DE1(G)α(n)(d+Δ-r-δ)2+[ζ(G)+2m]2n.

Since equality in 2.4 holds if and only if a1 = a2 = ⋯ = an and b1 = b2 = ⋯ = bn, the equality of the theorem holds if and only if ei +di is same for all the vertices of G.

Corollary 2.6

DE1(G)n2(d+Δ-r-δ)2+4[ζ(G)+2m]24n.

Proof

Since α(n)n24, the proof follows by above theorem.

Theorem 2.7

Let G be a nontrivial graph of order n and size m, then

DE1(G)(d+Δ+r+δ)[ζ(G)+2m]-(r+δ)(d+Δ)n.
Proof

Let a1, a2, …, an and b1, b2, …, bn be real numbers for which there exist real constants t and T, so that for each i, i = 1, 2, …, n, taibiTai. Then the following inequality is valid (see [5])

i=1nbi2+tTi=1nai2(t+T)i=1naibi

Equality of (2.5) holds if and only if, tai = bi = Tai for at least one i, 1 ≤ in We choose bi = ei + di, ai = 1, t = r + δ and T = d + Δ in inequality (2.5), then

i=1n(ei+di)2+(r+δ)(d+Δ)i=1n1(r+δ+d+Δ)i=1n(ei+di)DE1(G)+(r+δ)(d+Δ)n(r+δ+d+Δ)[ζ(G)+2m]DE1(G)(r+δ+d+Δ)[ζ(G)+2m]-(r+δ)(d+Δ)n.

If tai = bi = Tai for some i, then t = bi = T. Therefore equality holds if and only if r + δ = ei + di = d + Δ for some i. i.e., r = ei = d and δ = di = Δ for some i. Therefore equality of the theorem holds if and only if ei + di is same for all the vertices of G.

Lemma 2.8

([2]) For positive real numbers a1, a2,, ak

A12B1(k-1),

where

A=2k(k-1)(a1a2+a1a3++a1an+a2a3++ak-1ak),B=1k(a1a2ak-1+a1a2ak-2ak++a2a3ak-1ak).

Equality holds if and only if a1 = a2 = ⋯ = ak.

The Lagrange identity is as follows.

Lemma 2.9

([11]) Let (a) = (a1, a2,, ak), (b) = (b1, b2,, bk) be two real k–tuples. Then

i=1kai2j=1kbj2-(i=1kaibi)2=1i<jk(aibj-ajbi)2.

Theorem 2.10

Let G be a nontrivial graph of order n and size m, then

(i)E1(G)d2+[ζ(G)-d]2(n-1)+2(n-2)(n-1)2(d2-r)2

with equality if and only if G is a star,

(ii)E1(G)2d2+ζ2(G)-2dζ(G)-(n-1)(n-2)[1(n-1)dj=1nej[(i=1n1ei)-1d]]

with equality if and only if G is a star.

Proof

(i) If we set k = n − 1, ai = ei+1, bi = 1, i = 1, 2, …, k in Lemma 2.9 then we get

(n-1)i=2nei2-(i=2nei)2=2i<jn(ei-ej)2

If eiej, ij then e1 = d and hence

(n-1)[E1(G)-d2]=[ζ(G)-d]2+2i<jn(ei-ej)2

Now,

2i<jn[ei-ej]=(n-2)e2-i=3nei+3i<jn-1[ei-ej]+i=3n-1ei-(n-3)en=(n-2)(e2-en)+3i<jn-1[ei-ej](n-2)(d2-r).

By power-mean inequality [3], we have

(2i<jn(ei-ej)2(n-1)(n-2)2)122i<jn[ei-ej](n-1)(n-2)2

with equality if and only if

(ei-ej)=(el-em)

for any 2 ≤ i, j, l, mn, that is e2 = e3 = ⋯ = en−1 = en.

From the above, we get

2i<jn(ei-ej)22(n-1)(n-2)(2i<jn[ei-ej])2

with equality if and only if e2 = e3 = ⋯ = en−1 = en.

Using (2.9), from the above, we get

2i<jn(ei-ej)22(n-2)(n-1)(d2-r)2

Using the above result in (2.8), we get (2.6). First part of proof is done. Now, suppose that the equality holds in (2.6). Then the equality holds in (2.9) and (2.10). From the equality in (2.9), we get e3 = ⋯ = en−1. From the equality in (2.10), we get e2 = e3 = ⋯ = en−1 = en. Hence G is a star graph.

Conversely, suppose e2 = e3 = ⋯ = en−1 = en = r. Then we have

ζ(G)=d+(n-1)r

that is,

r=ζ(G)-d(n-1)

Using the above result we get

E1(G)=d2+(n-1)r2=d2+[ζ(G)-d]2(n-1)=d2+[ζ(G)-d]2(n-1)+2(n-2)(n-1)2(d2-r)2         as         d2=r

(ii) If we set k = n − 1, ai = ei+1, i = 1, 2, …, k in Lemma 2.8, then we get

2i<jneiej(n-1)(n-2)2[1n-1j-2neji=2n1ei]2n-2=(n-1)(n-2)2[1(n-1)dj=1nej[(i=1n1ei)-1d]]2n-2

But,

2i<jn(ei-ej)2=(n-2)i=2nei2-22i<jneiej(n-2)[E1(G)-d2]-(n-1)(n-2)[1(n-1)dj=1nej[(i=1n1ei)-1d]]2n-2

using the above result in (2.8), we get the upper bound in (2.7). First part of the proof is done.

The equality holds in (2.7) if and only if the equality holds in (2.11), that is, e2 = e3 = ⋯ = en−1 = en, by Lemma 2.8. Hence, the equality holds in (2.7) if and only if G is a star graph.

Conclusion

In this paper we have established some bounds of the first Zagreb eccentricity index and first Zagreb degree eccentricity index in terms of some graph parameters such as order, size, maximum and minimum degree, radius, diameter and total eccentricity index. It may be useful to give the bounds for E1(G), E2(G), DE1(G) and DE2(G) indices in terms of other graph invariants.

Acknowledgements

The first author is thankful to the University Grants Commission, Government of India, for the financial support under the Basic Science Research Fellowship. UGC vide No.F.25 – 1/2014 – 15(BSR)/7 – 349/2012(BSR), January 2015.

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