BOUNDS ON THE HYPER-ZAGREB INDEX

Abstract

The hyper-Zagreb index HM(G) of a simple graph G is defined as the sum of the terms (d_u+d_v)² over all edges uv of G, where d_u denotes the degree of the vertex u of G. In this paper, we present several upper and lower bounds on the hyper-Zagreb index in terms of some molecular structural parameters and relate this index to various well-known molecular descriptors.

1. Introduction

Let G be a finite simple connected graph with vertex set V (G) and edge set E(G). We denote by du the degree of the vertex u of G. A vertex u is said to be pendent if du = 1. We denote by δ and ∆ the minimal and maximal vertex degrees of G, respectively. The distance dG(u, v) between the vertices u and v of G is defined as the length of any shortest path in G connecting u and v. The eccentricity εu of a vertex u is the largest distance between u and any other vertex of G. For positive integers s ≠ t, a graph G is said to be (s, t)-semiregular if its vertex degrees assume only the values s and t, and if there is at least one vertex of degree s and at least one of degree t. A bipartite graph is said to be (s, t)-semiregular bipartite or (s, t)-biregular if any vertex in one side of the given bipartition has degree s and any vertex in the other side of the bipartition has degree t.

A molecular descriptor (also known as topological index or graph invariant) is any function on a graph that does not depend on a labeling of its vertices. In organic chemistry, topological indices have been found to be useful in chemical documentation, isomer discrimination, quantitative structure-property relationships (QSPR), quantitative structure-activity relationships (QSAR), and pharmaceutical drug design [5,12].

The Zagreb indices are among the oldest topological indices, and were introduced by Gutman and Trinajstić [13] in 1972. These indices have since been used to study molecular complexity, chirality, ZE–isomerism, and hetero–systems. The first and second Zagreb indices of G are denoted by M1(G) and M2(G), respectively, and defined as

The first Zagreb index can also be expressed as a sum over edges of G,

A multiplicative version of the first Zagreb index called multiplicative sum Zagreb index was proposed by Eliasi et al. [7] in 2010. The multiplicative sum Zagreb index of G is defined as

In 1975, Milan Randić [16] proposed a structural descriptor, based on the end-vertex degrees of edges in a graph, called the branching index that later became the well-known Randić connectivity index. The Randić index R(G) of G is defined as

The Randić index is one of the most successful molecular descriptors in QSPR and QSAR studies, suitable for measuring the extent of branching of the carbonatom skeleton of saturated hydrocarbons.

Another variant of the Randić connectivity index named the harmonic index was introduced by Fajtlowicz [8] in 1987. The harmonic index H(G) of G is defined as

Motivated by definition of the Randić connectivity index, Vukičević and Furtula [20] proposed another vertex-degree-based topological index, named the geometric-arithmetic index. The geometric-arithmetic index of a graph G is denoted by GA(G) and defined as

The eccentric connectivity index was introduced by Sharma et al. [17] in 1997. The eccentric connectivity index ξc(G) of G is defined as

The eccentric connectivity index can also be expressed as a sum over edges of G,

The Zagreb eccentricity indices were introduced by Vukičević and Graovac [21] in 2010. These indices are defined in analogy with the Zagreb indices by replacing the vertex degrees with the vertex eccentricities. Thus, the first and second Zagreb eccentricity indices of G are defined as

Recently, Shirdel et al. [18] introduced a variant of the first Zagreb index called hyper-Zagreb index. The hyper-Zagreb index of G is denoted by HM(G) and defined as

In this paper, we present several upper and lower bounds on the hyper-Zagreb index in terms of some graph parameters such as the order, size, number of pendant vertices, minimal and maximal vertex degrees, and minimal non-pendent vertex degree, and relate this index to various well-known graph invariants such as the first and second Zagreb indices, multiplicative sum Zagreb index, Randić index, harmonic index, geometric-arithmetic index, eccentric connectivity index, and second Zagreb connectivity index. We refer the reader to consult [1,2,3,6,9,10,11,19] for more information on computing bounds on vertex-degree-based topological indices.

 

2. Preliminaries

In this section, we recall some well-known inequalities which will be used throughout the paper.

Let x1, x2, . . . , xn be positive real numbers.

The arithmetic mean of x1, x2, . . . , xn is equal to

The geometric mean of x1, x2, . . . , xn is equal to

The harmonic mean of x1, x2, . . . , xn is equal to

Related to these three means, we have the following well-known inequalities.

Lemma 2.1 (AM-GM-HM inequality). Let x1, x2, . . . , xn be positive real numbers. Then

with equality if and only if x1 = x2 = . . . = xn.

Lemma 2.2 (Cauchy-Schwarz inequality). Let X = (x1, x2, . . . , xn) and Y = (y1, y2, . . . , yn) be two sequences of real numbers. Then

with equality if and only if the sequences X and Y are proportional, i.e., there exists a constant c such that xi = cyi, for each 1 ≤ i ≤ n.

As a special case of the Cauchy-Schwarz inequality, when y1 = y2 = · · · = yn, we get the following result.

Corollary 2.3. Let x1, x2, . . . , xn be real numbers. Then

with equality if and only if x1 = x2 = · · · = xn.

Lemma 2.4 (Pólya-Szegö inequality [15]). Let 0 < m1 ≤ xi ≤ M1 and 0 < m2 ≤ yi ≤ M2, for 1 ≤ i ≤ n. Then

Lemma 2.5 (Diaz-Metcalf inequality [4]). Let x1, x2, . . . , xn, y1, y2, . . . , yn be real numbers such that pxi ≤ yi ≤ Pxi, for 1 ≤ i ≤ n. Then

with equality if and only if yi = Pxi or yi = pxi, for 1 ≤ i ≤ n.

Lemma 2.6 ([14]). Let G be a nontrivial connected graph of order n. For each vertex u ∈ V (G),

with equality if and only if G≅ P4 or G ≅ Kn − iK2, , where P4 denotes the path on 4 vertices and Kn − iK2 denotes the graph obtained from the complete graph Kn by removing i independent edges.

 

3. Results and discussion

In this section, we present several upper and lower bounds on the hyper-Zagreb index in terms of some graph parameters and various molecular descriptors.

Throughout this section, we assume that G is a nontrivial simple connected graph with order n and size m. Note that, the connectivity of G is not an important restriction, since if G has connected components G1, G2, . . . , Gr, then . Furthermore, every molecular graph is connected.

Theorem 3.1. For any graph G,

with equality if and only if G is a regular graph.

Proof. Since 2δ ≤ du + dv ≤ 2∆, for each uv ∈ E(G), we have

The equalities hold if and only if du + dv = 2∆ = 2δ, for each uv ∈ E(G), which implies that G is a regular graph. □

Theorem 3.2. For any graph G with p pendant vertices and minimal non-pendent vertex degree δ1,

with equality if and only if G is regular or (1, ∆)-semiregular.

Proof. From the definition of the hyper-Zagreb index,

Similarly,

The above equalities hold if and only if du = dv = ∆ = δ1, for each uv ∈ E(G), with du, dv ≠ 1, and dv = ∆ = δ1, for each uv ∈ E(G), with du = 1. This implies that, G is (1, ∆)-semiregular if p > 0, and G is regular if p = 0. □

Theorem 3.3. Let G be a tree. Then

with equality if and only if G is a star graph.

Proof. Since du + dv ≤ n, for each uv ∈ E (G), we have

with equality if and only if du + dv = n, for each uv ∈ E(G), which implies that G is a star graph. □

Theorem 3.4. For any graph G,

with equality if and only if G is regular or biregular.

Proof. By Corollary 2.3, we obtain

The equality holds if and only if there exists a constant c such that du+dv = c, for each uv ∈ E(G). If uv, uz ∈ E(G), then du + dv = du + dz, which implies that dv = dz. Consequently, for each u ∈ V (G), every neighbor of u has the same degree. Since G is connected, this holds if and only if G is regular or biregular. □

Theorem 3.5. For any graph G,

Proof. Using the fact that, 2δ ≤ du + dv ≤ 2∆, for each uv ∈ E(G), and setting m1 = 2δ, xi = du+dv, 1 ≤ i ≤ m, M1 = 2∆, and m2 = yi = M2 = 1, 1 ≤ i ≤ m, in Pólya-Szegö inequality, we obtain

which is easily simplified into

Theorem 3.6. For any graph G,

with equality if and only if G is a regular graph.

Proof. By setting p = 2δ, P = 2∆, xi = 1, and yi = du + dv, 1 ≤ i ≤ m, in Diaz-Metcalf inequality, we obtain

which is easily simplified into

By Lemma 2.5, the equality holds if and only if du+dv = 2δ or du+dv = 2∆, for each uv ∈ E(G), which implies that G is a regular graph. □

Theorem 3.7. For any graph G,

with equality if and only if G is a regular graph.

Proof. Using the AM-GM inequality, we get

By Lemma 2.1, the equality holds if and only if du = dv, for each uv ∈ E (G), which implies that G is a regular graph. □

Theorem 3.8. For any graph G,

with equality if and only if G is a regular graph.

Proof. Using the definitions of the hyper-Zagreb and Zagreb indices, we have

Now using the fact that, δ ≤ du ≤ ∆, for each u ∈ V (G), we obtain

The equalities hold if and only if du = ∆ = δ, for each u ∈ V (G), which implies that G is a regular graph. □

Theorem 3.9. For any graph G,

with equality if and only if G is regular or biregular.

Proof. Using the AM-GM inequality, we get

By Lemma 2.1, the equality holds if and only if there exists a constant c such that (du + dv)2 = c, for each uv ∈ E(G). This implies that, , for each uv ∈ E(G). As explained in the proof of Theorem 3.4, this holds if and only if G is regular or biregular. □

Theorem 3.10. For any graph G,

with equality if and only if G is a regular graph.

Proof. It is easy to see that, for each uv ∈ E(G),

Now, from the definition of the hyper-Zagreb index,

The equalities hold if and only if du = dv = δ = ∆, for each uv ∈ E(G), which implies that G is a regular graph. □

Theorem 3.11. For any graph G,

with equality if and only if G is a regular graph.

Proof. Using the AM-HM inequality, AM-GM inequality, and Corollary 2.3, respectively, we obtain

By Lemma 2.1, the above first equality holds if and only if there exists a constant c such that , for each uv ∈ E(G). If uv, uz ∈ E(G), then , which implies that dv = dz. Consequently, for each vertex u ∈ V (G), every neighbor of u has the same degree. This holds if and only if G is regular or biregular. By Lemma 2.1, the second equality holds if and only if du = dv , for each uv ∈ E(G), which implies that G is a regular graph. By Corollary 2.3, the third equality holds if and only if there exists a constant c such that , or equivalently, du + dv = 2c, for each uv ∈ E(G). As explained in the proof of Theorem 3.4, this holds if and only if G is regular or biregular. Consequently, , with equality if and only if G is a regular graph. □

Theorem 3.12. For any graph G,

with equality if and only if G is a regular graph.

Proof. It is easy to see that, for each uv ∈ E(G),

Now, from the definition of the hyper-Zagreb index,

The equalities hold if and only if du = dv = δ = ∆, for each uv ∈ E(G), which implies that G is a regular graph. □

Theorem 3.13. For any graph G,

with equality if and only if G is a regular graph.

Proof. Using the Cauchy-Schwartz inequality, we get

By Lemma 2.2, the above first equality holds if and only if there exists a constant c such that , for each uv ∈ E(G). This implies that (du + dv)3 = 2c2, for each uv ∈ E(G). If uv, uz ∈ E(G), then (du + dv)3 = (du + dz)3, which is then easily simplified into dv = dz. Consequently, for each vertex u ∈ V (G), every neighbor of u has the same degree. This holds if and only if G is regular or biregular. The second equality holds if and only if du = dv = δ, for each uv ∈ E (G), which implies that G is a regular graph. Consequently, , with equality if and only if G is a regular graph. □

Theorem 3.14. For any graph G,

with equality if and only if G is a regular graph.

Proof. Using the AM-HM inequality and Corollary 2.3, we obtain

By Lemma 2.1, the above first equality holds if and only if du = dv, for each uv ∈ E(G), which implies that G is a regular graph. By Corollary 2.3, the second equality holds if and only if there exists a constant c such that , for each uv ∈ E(G). If uv, uz ∈ E(G), then . Then dv (du + dz) = dz(du + dv), which is easily simplified into dv = dz. So, every neighbor of u has the same degree, which implies that G is regular or biregular. The third equality holds if and only if du = dv = δ, for each uv ∈ E(G), which implies that G is a regular graph. Consequently, , with equality if and only if G is a regular graph. □

Theorem 3.15. For any graph G,

with equality if and only if G is a regular graph.

Proof. Using the AM-HM inequality, we obtain

By Lemma 2.1, the above first equality holds if and only if there exists a constant c such that , for each uv ∈ E(G). Using the same argument as in the previous theorems, this holds if and only if G is regular or biregular. The second equality holds if and only if du = dv = δ, for each uv ∈ E(G), which implies that G is a regular graph. Consequently, , with equality if and only if G is a regular graph. □

Theorem 3.16. For any graph G,

where ξ3(G) = ∑u∈V(G) (εu2 + εv2), and the equality holds if and only if G≅ P4 or G ≅ Kn − iK2,

Proof. Using the definition of the hyper-Zagreb index and Lemma 2.6, we get

By Lemma 2.6, the equality holds if and only if du = n−εu, for each u ∈ V (G), which by Lemma 2.6 implies that, G≅ P4 or G ≅ Kn − iK2, □