Javaid et al. Journal of Inequalities and Applications (2019) 2019:37 https://doi.org/10.1186/s13660-019-1990-1

R E S E A R C H Open Access

On some bounds of the topological indicesof generalized Sierpinski and extendedSierpinski graphsImran Javaid1* , Hira Benish1, Muhammad Imran2,3, Amna Khan1 and Zafar Ullah4

*Correspondence:imran.javaid@bzu.edu.pk1Centre for Advanced Studies inPure and Applied Mathematics,Bahauddin Zakariya University,Multan, PakistanFull list of author information isavailable at the end of the article

AbstractSierpinski graphs are extensively studied graphs of fractal nature with applications intopology, mathematics of Tower of Hanoi and computer science. The generalizedSierpinski graphs are defined by replication of exactly the same graph, yieldingself-similar graph. Certain graph invariants referred to as topological indices are usedto determine a large number of properties like physico-chemical properties,thermodynamic properties, chemical activity and biological activity of chemicalgraphs. In QSAR/QSPR study, these graph invariants play a vital role.In this article, we study the topological indices of generalized Sierpinski and

extended Sierpinski graphs with an arbitrary base graph. We obtain bounds for theatom-bond connectivity index, harmonic index, Zagreb indices and sum-connectivityindex for the generalized Sierpinski graphs and extended Sierpinski graphs.

MSC: 05C12; 05C70; 05C76

Keywords: Atom-bond connectivity index; Harmonic index; Zagreb indices;Sum-connectivity index; Generalized Sierpinski network; Extended Sierpinski network

1 IntroductionApplications of molecular structure descriptors are a standard procedure in the study ofstructure–property relations nowadays, especially in the field of QSAR/QSPR study. Dur-ing the last century, theoretical chemists started working on the use of topological indicesto obtain information of various properties of organic substances which depend upontheir molecular structure. For this purpose, numerous topological indices were found andstudied in the chemical literature [28]. Todeschini et al. used two zeroth-order and twofirst-order connectivity indices for the first time as descriptors in structure–property cor-relations in an optimization study. A set of new formulas for heat capacity, glass transitiontemperature, refractive index, cohesive energy and dielectric constant were introducedthat were based on these descriptors. The Randić index has been used to parallel the boil-ing point and Kovats constants, and was closely correlated with many chemical properties.

A graph invariant that correlates the physico-chemical properties of a molecular graphwith a number is called a topological index [18]. The first topological index was intro-duced by Wiener, a chemist, in 1947 to calculate the boiling points of paraffins in [29]. Za-greb indices, derived in 1972 by Gutman and Trinajstic [17], are used to study molecules

© The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.

Javaid et al. Journal of Inequalities and Applications (2019) 2019:37 Page 2 of 14

and complexity of selected classes of molecules. Zagreb indices have found an interestinguse in the QSPR/QSAR modeling and are useful in the study of anti-inflammatory activ-ities of certain chemical instances. The harmonic index is one of the variants of degree-based topological indices which has also been studied in relationship with eigenvalues byFavaron et al. in [9]. The atom-bond connectivity (ABC) index, introduced by Estrada etal. [4], gives a good model for the stability of linear and branched alkanes as well as thestrain energy of cycloalkanes. The ABC index is used to predict the bioactivity of chemicalcompounds. The sum-connectivity index was proposed by Zhou et al. [31] and studied inrelationship with the Randić index in [1]. Many topological indices have been studied inthe literature [3–27].

For the purpose of this article, we use the following standard notations: Given a graphG = (V , E) and for u, v ∈ V , u being adjacent to v is denoted by u ∼ v and {u, v} ∈ E. The dis-tance between any two vertices u and v of G is the minimum number of edges in the short-est u– v path and is denoted by d(u, v). The eccentricity ecc(v) of a vertex v in G is the max-imum distance between v and any of the other vertex of G, i.e. ecc(v) = maxu∈V (G) d(v, u).The diameter diam(G) of G is the diam(G) = maxv∈V (G) ecc(v) and the radius rad(G) of Gis the rad(G) = minv∈V (G) ecc(v). The number of adjacent vertices of v is called the degreeof v in G, denoted by dG(v). The maximum and minimum vertex degree in a graph G, aredefined as �(G) = max{dG(u) : u ∈ V (G)} and δ(G) = min{dG(u) : u ∈ V (G)}, respectively.

The generalized Sierpiński graph S(G, t) is a graph with V t is the vertex set of S(G, t)and V = V (G). The vertex set V t is the set of all words x1x2...xt of length t where xr ∈ V ,1 ≤ r ≤ t. {x, y} ∈ E(S(G, t)) if and only if there exists i ∈ {1, . . . , n} such that:

1. xj = yj, if j < i;2. xi �= yi and {xi, yi} ∈ E(G);3. xj = yi and yj = xi if j > i.

The S(Kn, 3) are Tower of Hanoi graphs. The Sierpiński graph is a graph with G = Kn,introduced in [23, 24]. Gravier et al. constructed the generalized Sierpiński graphs S(G, t)in [15].

Notice that if {x, y} is an edge of S(G, t) then there is an edge {u, v} of G and a word w suchthat x = wuvv · · · v and y = wvuu · · ·u. Extreme vertices are the vertices of the form uu · · ·u.Note that, for any graph G of order n and any integer t ≥ 2, S(G, t) has n extreme verticesand, if u has degree dG(u), then the extreme vertex uu · · ·u of S(G, t) also has degree dG(u).The degrees of vertices of the form vuu · · ·u and uvv · · · v are equal to dG(u) + 1 and dG(v) +1, respectively. For better understanding of S(G, t), please see Fig. 1.

It is clear that u as a vertex of V (S(G, t)) has degree dS(G,t)(u) ∈ {dG(u), dG(u) + 1} wheredG(u) is degree of u in G. We use the terminology of [5]. �S(G,t)(dG(u), dG(v)) is the numberof copies of {u, v} edge with degrees dG(u) and dG(v) in S(G, t). N(u) = {s ∈ V : {u, s} ∈ E}is the neighborhood of u ∈ V . For u, v ∈ V , the number of triangles of G containing u andv will be denoted by τ (u, v) and the number of triangles of G will be denoted by τ (G).For any pair of adjacent vertices u, v ∈ V , we have |N(u) ∩ N(v)| = τ (u, v), |N(u) ∪ N(v)| =dG(u) + dG(v) – τ (u, v) and |N(u) – N(v)| = dG(u) – τ (u, v). From now onward, for a graphof order n, we will use the function ψn(t) = 1 + n + n2 + · · · + nt–1 = nt–1

n–1 .

Lemma 1 ([5]) For any integer t ≥ 2 and any edge {u, v} of a graph G of order n,1. �S(G,t)(dG(u), dG(v)) = nt–2(n – dG(u) – dG(v) + τ (u, v)).2. �S(G,t)(dG(u), dG(v) + 1) = nt–2(dG(v) – τ (u, v)) – ψn(t – 2)dG(u).

Javaid et al. Journal of Inequalities and Applications (2019) 2019:37 Page 3 of 14

Figure 1 Generalized Sierpinski Graphs S(1,C4) and S(3,C4)

3. �S(G,t)(dG(u) + 1, dG(v)) = nt–2(dG(u) – τ (u, v)) – ψn(t – 2)dG(v).4. �S(G,t)(dG(u) + 1, dG(v) + 1) = nt–2(τ (u, v) + 1) + ψn(t – 2)(dG(u) + dG(v) + 1).

Sierpiński graphs S(Kn, t) originated from the topological study of the Lipscomb space.WK-recursive network is a class of graphs, which was introduced in computer science in[2, 11]. WK-recursive networks are structurally close to Sierpiński graphs, these can be ob-tained from Sierpiński graphs by adding an open edge to each of its extreme vertices. Thegraphs S(G, t) have been studied from different points of views. Colorings of these graphshave been found in [10, 21], crossing number and several metric invariants such as unique1-perfect codes, average distance of these graphs have been studied in the literature. Formore literature on Sierpiński graphs, please see [14, 19, 20, 24, 30].

In this paper, we study the topological indices of generalized Sierpiński and extendedSierpiński graphs with an arbitrary base graph. Bounds for the atom-bond connectivityindex, harmonic index, Zagreb indices and sum-connectivity index for the generalizedSierpiński graphs and extended Sierpiński graphs have been determined. The atom-bondconnectivity index, harmonic index, Zagreb indices and sum-connectivity index are thevertex-degree-based topological indices that can be studied in terms of topological indicesof the base graph of a generalized Sierpiński graph, therefore we choose these indices forthe study of topological indices of generalized Sierpiński and extended Sierpiński graphs.

2 Topological indices of generalized Sierpinski graphsThe atom-bond connectivity (ABC) index, introduced by Estrada et al. [4], a degree basedindex, is defined as

ABC(G) =∑

u∼v

√dG(u) + dG(v) – 2

dG(u)dG(v).

Javaid et al. Journal of Inequalities and Applications (2019) 2019:37 Page 4 of 14

In the following theorem, we present a lower bound for the atom-bond connectivity indexfor generalized Sierpiński graphs in terms of the atom-bond connectivity index of G.

Theorem 1 For any graph G of order n ≥ 3, δ(G) ≥ 2 and any integer t ≥ 2,

ABC(S(G, t)

) ≤ ψn(t) ABC(G).

Proof The ABC index of S(G, t) can be expressed as

ABC(S(G, t)

)=

∑

u∼v

1∑

i=0

1∑

j=0

�S(G,t)(dG(u) + i, dG(v) + j

)√

(dG(u) + i) + (dG(v) + j) – 2(dG(u) + i)(dG(v) + j)

by using Lemma 1:

=∑

u∼v

{nt–2(n – dG(u) – dG(v) + τ (u, v)

)√

dG(u) + dG(v) – 2dG(u)dG(v)

+ nt–2(dG(v) – τ (u, v))

– ψn(t – 2)dG(u)

√dG(u) + dG(v) – 1dG(u)(dG(v) + 1)

+ nt–2(dG(u) – τ (u, v))

– ψn(t – 2)dG(v)

√dG(u) + dG(v) – 1(dG(u) + 1)dG(v)

+ nt–2(τ (u, v) + 1)

+ ψn(t – 2)(dG(u) + dG(v) + 1

)√

dG(u) + dG(v)(dG(u) + 1)(dG(v) + 1)

}.

We have δ(G) ≥ 2,√

dG(u)+dG(v)–1dG(u)(dG(v)+1) ≤

√dG(u)+dG(v)–2

dG(u)dG(v) ,√

dG(u)+dG(v)–1(dG(u)+1)dG(v) ≤

√dG(u)+dG(v)–2

dG(u)dG(v) and√

dG(u)+dG(v)(dG(u)+1)(dG(v)+1) ≤

√dG(u)+dG(v)–2

dG(u)dG(v) . We deduce

ABC(S(G, t)

)

≤∑

u∼v

{nt–2(n – dG(u) – dG(v) + τ (u, v)

)+ nt–2(dG(v) – τ (u, v)

)

– ψn(t – 2)dG(u) + nt–2(dG(u) – τ (u, v))

– ψn(t – 2)dG(v)

+ nt–2(τ (u, v) + 1)

+ ψn(t – 2)(dG(u) + dG(v) + 1

)}√

dG(u) + dG(v) – 2dG(u)dG(v)

=∑

u∼v

{nt–1 + nt–2 + ψn(t – 2)

}√

dG(u) + dG(v) – 2dG(u)dG(v)

=∑

u∼vψn(t)

√dG(u) + dG(v) – 2

dG(u)dG(v),

which gives the desired result:

ABC(S(G, t)

) ≤ ψn(t) ABC(G). �

Javaid et al. Journal of Inequalities and Applications (2019) 2019:37 Page 5 of 14

The harmonic index, another variant of degree based topological indices, is defined asfollows:

H(G) =∑

u∼v

2dG(u) + dG(v)

.

Now, we compute a lower bound for the harmonic index for generalized Sierpiński graphs.

Theorem 2 For any graph G of order n ≥ 2, size m ≥ 1 and any integer t ≥ 2,

H(S(G, t)

) ≤ ψn(t)H(G).

Proof The H(G) index of S(G, t) can be expressed as

H(S(G, t)

)=

∑

u∼v

1∑

i=0

1∑

j=0

�S(G,t)(dG(u) + i, dG(v) + j

) 2(dG(u) + i) + (dG(v) + j)

.

By using Lemma 1 and 2dG(u)+dG(v)+j ≤ 2

dG(u)+dG(v) , for j = 1, 2. We have

H(S(G, t)

) ≤∑

u∼v

{nt–2(n – dG(u) – dG(v) + τ (u, v)

)+ nt–2(dG(v) – τ (u, v)

)

– ψn(t – 2)dG(u) + nt–2(dG(u) – τ (u, v))

– ψn(t – 2)dG(v)

+ nt–2(τ (u, v) + 1)

+ ψn(t – 2)(dG(u) + dG(v) + 1

)} 2dG(u) + dG(v)

=∑

u∼v

{nt–1 + nt–2 + ψn(t – 2)

} 2dG(u) + dG(v)

=∑

u∼vψn(t)

2dG(u) + dG(v)

.

We get the required result:

H(S(G, t)

) ≤ ψn(t)H(G). �

Bounds on the Zagreb first and second indices, defined as

Z1(G) =∑

u∼v

(dG(u) + dG(v)

)

and

Z2(G) =∑

u∼v

(dG(u) · dG(v)

)

for generalized Sierpiński graphs are presented in the following theorems.

Theorem 3 For any graph G of order n ≥ 2, size m ≥ 1 and any integer t ≥ 2,

Z1(S(G, t)

) ≤ 2mψn(t)(n – rad(G) + 1

).

Javaid et al. Journal of Inequalities and Applications (2019) 2019:37 Page 6 of 14

Proof The Z1(G) index of S(G, t) can be expressed as

Z1(S(G, t)

)=

∑

u∼v

1∑

i=0

1∑

j=0

�S(G,t)(dG(u) + i, dG(v) + j

)((dG(u) + i

)+

(dG(v) + j

)).

It is clear that dG(u) + dG(v) ≤ dG(u) + dG(v) + 1 ≤ dG(u) + dG(v) + 2. Using Lemma 1 andthe above inequality we get

Z1(S(G, t)

) ≤∑

u∼v

{nt–2(n – dG(u) – dG(v) + τ (u, v)

)

+ nt–2(dG(v) – τ (u, v))

– ψn(t – 2)dG(u) + nt–2(dG(u) – τ (u, v))

– ψn(t – 2)dG(v) + nt–2(τ (u, v) + 1)

+ ψn(t – 2)(dG(u) + dG(v) + 1

)}(dG(u) + dG(v) + 2

)

=∑

u∼v

{nt–1 + nt–2 + ψn(t – 2)

}(dG(u) + dG(v) + 2

)

=∑

u∼vψn(t)

(dG(u) + dG(v) + 2

).

By using dG(u) ≤ n – ecc(u) and rad(G) ≤ ecc(u), we get

dG(u) + dG(v) + 2 ≤ 2(n – rad(G) + 1

).

Using the above inequality, we get the required result:

Z1(S(G, t)

) ≤ 2mψn(t)(n – rad(G) + 1

). �

Theorem 4 For any graph G of order n ≥ 2, size m ≥ 1 and any integer t ≥ 2,

Z2(S(G, t)

) ≤ mψn(t)(n – rad(G) + 1

)2.

Proof The Z2(G) index of S(G, t) can be expressed as

Z2(S(G, t)

)=

∑

u∼v

1∑

i=0

1∑

j=0

�S(G,t)(dG(u) + i, dG(v) + j

)((dG(u) + i

) · (dG(v) + j))

.

It is clear that dG(u) · dG(v) ≤ dG(u) · (dG(v) + 1) ≤ (dG(u) + 1) · dG(v) ≤ (dG(u) + 1) ·(dG(v) + 1). Using Lemma 1 and the above inequality we get

Z2(S(G, t)

)

≤∑

u∼v

{nt–2(n – dG(u) – dG(v) + τ (u, v)

)+ nt–2(dG(v) – τ (u, v)

)

– ψn(t – 2)dG(u) + nt–2(dG(u) – τ (u, v))

– ψn(t – 2)dG(v)

+ nt–2(τ (u, v) + 1)

+ ψn(t – 2)(dG(u) + dG(v) + 1

)}(dG(u) + 1

) · (dG(v) + 1)

Javaid et al. Journal of Inequalities and Applications (2019) 2019:37 Page 7 of 14

=∑

u∼v

{nt–1 + nt–2 + ψn(t – 2)

}(dG(u) + 1

) · (dG(v) + 1)

=∑

u∼vψn(t)

(dG(u) + 1

) · (dG(v) + 1).

We have (dG(u) + 1) · (dG(v) + 1) ≤ (n – rad(G) + 1)2. Using the above inequality, we get therequired result:

Z2(S(G, t)

) ≤ mψn(t)(n – rad(G) + 1

)2. �

The sum-connectivity index was proposed by Zhou et al. [31]. The sum-connectivityindex χ (G) is defined as

χ (G) =∑

u∼v

1√dG(u) + dG(v)

.

In the following result, we discuss the sum-connectivity index of generalized Sierpińskigraphs.

Theorem 5 For any graph G of order n ≥ 2, size m ≥ 1 and any integer t ≥ 2,

χ(S(G, t)

) ≤ ψn(t)χ (G).

Proof The χ (S(G, t)) index of S(G, t) can be expressed as

χ(S(G, t)

)=

∑

u∼v

1∑

i=0

1∑

j=0

�S(G,t)(dG(u) + i, dG(v) + j

) 1(dG(u) + i) + (dG(v) + j)

.

We have 1√dG(u)+dG(v)+1

≤ 1√dG(u)+dG(v)

and 1√dG(u)+dG(v)+2

≤ 1√dG(u)+dG(v)

. By using Lemma 1

and these inequalities, we have

χ(S(G, t)

) ≤∑

u∼v

{nt–2(n – dG(u) – dG(v) + τ (u, v)

)+ nt–2(dG(v) – τ (u, v)

)

– ψn(t – 2)dG(u) + nt–2(dG(u) – τ (u, v))

– ψn(t – 2)dG(v)

+ nt–2(τ (u, v) + 1)

+ ψn(t – 2)(dG(u) + dG(v) + 1

)} 1√dG(u) + dG(v)

=∑

u∼v

{nt–1 + nt–2 + ψn(t – 2)

} 1√dG(u) + dG(v)

=∑

u∼vψn(t)

1√dG(u) + dG(v)

,

which gives the desired result:

χ(S(G, t)

) ≤ ψn(t)χ (G). �

Javaid et al. Journal of Inequalities and Applications (2019) 2019:37 Page 8 of 14

Figure 2 Extended Sierpinski graph

3 Topological indices of extended Sierpinski graphsThe extended Sierpiński graph S(G, t+) was introduced by Klavžar et al. in [25]. The ex-tended Sierpiński graph S(G, t+) is obtained from S(G, t) by adding a new vertex x, calledthe special vertex of S(G, t+), and edges joining x with all extreme vertices � of S(G, t),as shown in Fig. 2. These edges are dS(G,t)(s) ∈ {dG(s), dG(s) + 1}. The number of extremevertices in S(G, t) is always the order of G. Therefore, in analogy to Lemma 1, we give thefollowing result for extended Sierpiński graphs.

Lemma 2 For any integer t ≥ 2 and any edge {u, v} of a graph G of order n.1. �S(G,t+)(dG(u), dG(v)) = nt–2(n – dG(u) – dG(v) + τ (u, v)).2. �S(G,t+)(dG(u), dG(v) + 1) = nt–2(dG(v) – τ (u, v)) – ψn(t – 2)dG(u).3. �S(G,t+)(dG(u) + 1, dG(v)) = nt–2(dG(u) – τ (u, v)) – ψn(t – 2)dG(v).4. �S(G,t+)(dG(u) + 1, dG(v)) = nt–2(τ (u, v) + 1) + ψn(t – 2)(dG(u) + dG(v) + 1).5. �S(G,t+)(dG(� ), dS(G,t+)(x)) = n.

We use the above lemma to find topological indices for extended Sierpiński graphsS(G, t+).

Theorem 6 For any graph G of order n ≥ 3, δ(G) ≥ 2 and any integer t ≥ 2,

ABC(S(G, t+)) ≤ (

n + ψn(t))

ABC(G).

Javaid et al. Journal of Inequalities and Applications (2019) 2019:37 Page 9 of 14

Proof The ABC index of S(G, t+) can be expressed as

ABC(S(G, t+))

=∑

u∼v

1∑

i=0

1∑

j=0

�S(G,t+)(dG(u) + i, dG(v) + j

)√

(dG(u) + i) + (dG(v) + j) – 2(dG(u) + i)(dG(v) + j)

+ �S(G,t+)(dG(� ), dS(G,t+)(x)

)√

dG(� ) + dS(G,t+)(x) – 2dG(� )dS(G,t+)(x)

;

by using Lemma 2

ABC(S(G, t+))

=∑

u∼v

{nt–2(n – dG(u) – dG(v) + τ (u, v)

)√

dG(u) + dG(v) – 2dG(u)dG(v)

+ nt–2(dG(v) – τ (u, v))

– ψn(t – 2)dG(u)

√dG(u) + dG(v) – 1dG(u)(dG(v) + 1)

+ nt–2(dG(u) – τ (u, v))

– ψn(t – 2)dG(v)

√dG(u) + dG(v) – 1(dG(u) + 1)dG(v)

+ nt–2(τ (u, v) + 1)

+ ψn(t – 2)(dG(u) + dG(v) + 1

)√

dG(u) + dG(v)(dG(u) + 1)(dG(v) + 1)

+ n

√dG(� ) + dS(G,t+)(x) – 2

dG(� )dS(G,t+)(x)

}.

We have δ(G) ≥ 2,√

dG(u)+dG(v)–1dG(u)(dG(v)+1) ≤

√dG(u)+dG(v)–2

dG(u)dG(v) ,√

dG(u)+dG(v)–1(dG(u)+1)dG(v) ≤

√dG(u)+dG(v)–2

dG(u)dG(v) ,√

dG(u)+dG(v)(dG(u)+1)(dG(v)+1) ≤

√dG(u)+dG(v)–2

dG(u)dG(v) and√

dG(� )+dS(G,t+)(x)–2dG(� )dS(G,t+)(x) ≤

√dG(u)+dG(v)–2

dG(u)dG(v) . We have

ABC(S(G, t+))

≤∑

u∼v

{nt–2(n – dG(u) – dG(v) + τ (u, v)

)+ nt–2(dG(v) – τ (u, v)

)

– ψn(t – 2)dG(u) + nt–2(dG(u) – τ (u, v))

– ψn(t – 2)dG(v)

+ nt–2(τ (u, v) + 1)

+ ψn(t – 2)(dG(u) + dG(v) + 1

)+ n

}√

dG(u) + dG(v) – 2dG(u)dG(v)

=∑

u∼v

{nt–1 + nt–2 + ψn(t – 2) + n

}√

dG(u) + dG(v) – 2dG(u)dG(v)

=∑

u∼v

(n + ψn(t)

)√

dG(u) + dG(v) – 2dG(u)dG(v)

and we get the required result:

ABC(S(G, t+)) ≤ (

n + ψn(t))

ABC(G). �

Javaid et al. Journal of Inequalities and Applications (2019) 2019:37 Page 10 of 14

A lower bound for the harmonic index of extended Sierpiński graphs S(G, t+) is given inthe following result.

Theorem 7 For any graph G of order n ≥ 2, size m ≥ 1 and any integer t ≥ 2,

H(S(G, t+)) ≤ (

n + ψn(t))H(G).

Proof The H(G) index of S(G, t+) can be expressed as

H(S(G, t+))

=∑

u∼v

1∑

i=0

1∑

j=0

�S(G,t+)(dG(u) + i, dG(v) + j

) 2(dG(u) + i) + (dG(v) + j)

+ �S(G,t+)(dG(� ), dS(G,t+)(x)

) 2dG(� ) + dS(G,t+)(x)

.

We have 2dG(u)+dG(v)+j ≤ 2

dG(u)+dG(v) , for j = 1, 2 and 2dG(� )+dS(G,t+)(x) ≤ 2

dG(u)+dG(v) . By usingLemma 2 and the above inequalities, we have

H(S(G, t+)) ≤

∑

u∼v

{nt–2(n – dG(u) – dG(v) + τ (u, v)

)+ nt–2(dG(v) – τ (u, v)

)

– ψn(t – 2)dG(u) + nt–2(dG(u) – τ (u, v))

– ψn(t – 2)dG(v)

+ nt–2(τ (u, v) + 1)

+ ψn(t – 2)(dG(u) + dG(v) + 1

)+ n

} 2dG(u) + dG(v)

=∑

u∼v

{nt–1 + nt–2 + ψn(t – 2) + n

} 2dG(u) + dG(v)

=∑

u∼v

(n + ψn(t)

) 2dG(u) + dG(v)

,

which gives the desired result:

H(S(G, t+)) ≤ (

n + ψn(t))H(G). �

Now, we discuss the Zagreb indices of extended Sierpiński graphs S(G, t+).

Theorem 8 For any graph G of order n ≥ 2, size m ≥ 1 and any integer t ≥ 2,

Z1(S(G, t+)) ≤ 2m

(n + ψn(t)

)(n – rad(G) + 1

).

Proof The Z1 index of S(G, t+) can be expressed as

Z1(S(G, t+))

=∑

u∼v

1∑

i=0

1∑

j=0

�S(G,t)(dG(u) + i, dG(v) + j

)((dG(u) + i

)+

(dG(v) + j

))

+ �S(G,t+)(dG(� ), dS(G,t+)(x)

)(dG(� ) + dS(G,t+)(x)

).

Javaid et al. Journal of Inequalities and Applications (2019) 2019:37 Page 11 of 14

It is clear that dG(u) + dG(v) ≤ dG(u) + dG(v) + 1 ≤ dG(u) + dG(v) + 2 and similarly (dG(� ) +dS(G,t+)(x)) ≤ dG(u) + dG(v) + 2. Using Lemma 2 and the above inequalities we get

Z1(S(G, t+))

≤∑

u∼v

{nt–2(n – dG(u) – dG(v) + τ (u, v)

)+ nt–2(dG(v) – τ (u, v)

)

– ψn(t – 2)dG(u) + nt–2(dG(u) – τ (u, v))

– ψn(t – 2)dG(v)

+ nt–2(τ (u, v) + 1)

+ ψn(t – 2)(dG(u) + dG(v) + 1

)+ n

}(dG(u) + dG(v) + 2

)

=∑

u∼v

{nt–1 + nt–2 + ψn(t – 2) + n

}(dG(u) + dG(v) + 2

)

=∑

u∼v

(n + ψn(t)

)(dG(u) + dG(v) + 2

).

We have dG(u) + dG(v) + 2 ≤ 2(n – rad(G) + 1) Using the above inequality, we get the re-quired result:

Z1(S(G, t+)) ≤ 2m

(n + ψn(t)

)(n – rad(G) + 1

). �

Theorem 9 For any graph G of order n ≥ 2, size m ≥ 1 and any integer t ≥ 2,

Z2(S(G, t+)) ≤ m

(n + ψn(t)

)(n – rad(G) + 1

)2.

Proof The Z2 index of S(G, t+) can be expressed as

Z2(S(G, t+))

=∑

u∼v

1∑

i=0

1∑

j=0

�S(G,t+)(dG(u) + i, dG(v) + j

)((dG(u) + i

) · (dG(v) + j))

+ �S(G,t+)(dG(� ), dS(G,t+)(x)

)(dG(� )

) · (dS(G,t+)(x)).

It is clear that dG(u) · dG(v) ≤ dG(u) · (dG(v) + 1) ≤ (dG(u) + 1) · dG(v) ≤ (dG(u) + 1) · (dG(v) +1) and ((dG(� )) · (dS(G,t+)(x))) ≤ (dG(u) + 1) · (dG(v) + 1). Using Lemma 2 and the aboveinequalities, we get

Z2(S(G, t+)) ≤

∑

u∼v

{nt–2(n – dG(u) – dG(v) + τ (u, v)

)

+ nt–2(dG(v) – τ (u, v))

– ψn(t – 2)dG(u) + nt–2(dG(u) – τ (u, v))

– ψn(t – 2)dG(v) + nt–2(τ (u, v) + 1)

+ ψn(t – 2)(dG(u) + dG(v) + 1

)+ n

}(dG(u) + 1

) · (dG(v) + 1)

=∑

u∼v

{nt–1 + nt–2 + ψn(t – 2) + n

}(dG(u) + 1

) · (dG(v) + 1)

=∑

u∼v

(n + ψn(t)

)(dG(u) + 1

) · (dG(v) + 1).

Javaid et al. Journal of Inequalities and Applications (2019) 2019:37 Page 12 of 14

We have (dG(u) + 1) · (dG(v) + 1) ≤ (n – rad(G) + 1)2. Using this inequality, we get the re-quired result:

Z2(S(G, t+)) ≤ m

(n + ψn(t)

)(n – rad(G) + 1

)2. �

The sum-connectivity index of extended Sierpiński graphs S(G, t+) is calculated in thefollowing result.

Theorem 10 For any graph G of order n ≥ 2, size m ≥ 1 and any integer t ≥ 2,

χ(S(G, t+)) ≤ (

n + ψn(t))χ (G).

Proof The χ (S(G, t+)) index of S(G, t+) can be expressed as

χ(S(G, t+))

=∑

u∼v

1∑

i=0

1∑

j=0

�S(G,t)(dG(u) + i, dG(v) + j

) 1√(dG(u) + i) + (dG(v) + j)

+ �S(G,t)(dG(� ), dS(G,t+)(x)

) 1√dG(� ) + dS(G,t+)(x)

.

We have 1√dG(u)+dG(v)+1

≤ 1√dG(u)+dG(v)

, 1√dG(u)+dG(v)+2

≤ 1√dG(u)+dG(v)

and 1√dG(� )+dS(G,t+)(x)

≤1√

dG(u)+dG(v). By using Lemma 2 and these inequalities we have

χ(S(G, t+)) ≤

∑

u∼v

{nt–2(n – dG(u) – dG(v) + τ (u, v)

)+ nt–2(dG(v) – τ (u, v)

)

– ψn(t – 2)dG(u) + nt–2(dG(u) – τ (u, v))

– ψn(t – 2)dG(v)

+ nt–2(τ (u, v) + 1)

+ ψn(t – 2)(dG(u) + dG(v) + 1

)+ n

} 1√dG(u) + dG(v)

=∑

u∼v

{nt–1 + nt–2 + ψn(t – 2) + n

} 1√dG(u) + dG(v)

=∑

u∼v

(n + ψn(t)

) 1√dG(u) + dG(v)

,

which gives the desired result:

χ(S(G, t+)) ≤ (

n + ψn(t))χ (G). �

4 ConclusionIn this paper, we have studied the topological indices of generalized Sierpiński and ex-tended Sierpiński graphs with an arbitrary base graph. We have obtained some upperbounds in terms of some standard graph-theoretic parameters like order, size, radius andin terms of the topological indices of the base graph G. We have determined bounds forthe atom-bond connectivity index, harmonic index, Zagreb indices and sum-connectivityindex for the generalized Sierpiński graphs and extended Sierpiński graphs.

AcknowledgementsNot applicable.

Javaid et al. Journal of Inequalities and Applications (2019) 2019:37 Page 13 of 14

FundingThis research is supported by the start up research grant 2016 of United Arab Emirates University, Al Ain, United ArabEmirates, via Grant No. G00002233.

Availability of data and materialsNot applicable.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors contributed equally and significantly in conducting this research work and writing this paper. All authors readand approved the final manuscript.

Author details1Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan.2Department of Mathematical Sciences, United Arab Emirates University, Al Ain, United Arab Emirates. 3Department ofMathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad,Pakistan. 4Department of Mathematics, University of Education Lahore Campus Dera Ghazi Khan, Dera Ghazi Khan,Pakistan.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 12 September 2018 Accepted: 30 January 2019

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