Skip to main content

Advertisement

We're creating a new version of this page. See preview

  • Research
  • Open Access

On some bounds of the topological indices of generalized Sierpiński and extended Sierpiński graphs

Journal of Inequalities and Applications20192019:37

https://doi.org/10.1186/s13660-019-1990-1

  • Received: 12 September 2018
  • Accepted: 30 January 2019
  • Published:

Abstract

Sierpiński graphs are extensively studied graphs of fractal nature with applications in topology, mathematics of Tower of Hanoi and computer science. The generalized Sierpiński graphs are defined by replication of exactly the same graph, yielding self-similar graph. Certain graph invariants referred to as topological indices are used to determine a large number of properties like physico-chemical properties, thermodynamic properties, chemical activity and biological activity of chemical graphs. In QSAR/QSPR study, these graph invariants play a vital role.

In this article, we study the topological indices of generalized Sierpiński and extended Sierpiński graphs with an arbitrary base graph. We obtain bounds for the atom-bond connectivity index, harmonic index, Zagreb indices and sum-connectivity index for the generalized Sierpiński graphs and extended Sierpiński graphs.

Keywords

  • Atom-bond connectivity index
  • Harmonic index
  • Zagreb indices
  • Sum-connectivity index
  • Generalized Sierpiński network
  • Extended Sierpiński network

MSC

  • 05C12
  • 05C70
  • 05C76

1 Introduction

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

A graph invariant that correlates the physico-chemical properties of a molecular graph with a number is called a topological index [18]. The first topological index was introduced by Wiener, a chemist, in 1947 to calculate the boiling points of paraffins in [29]. Zagreb indices, derived in 1972 by Gutman and Trinajstic [17], are used to study molecules and complexity of selected classes of molecules. Zagreb indices have found an interesting use in the QSPR/QSAR modeling and are useful in the study of anti-inflammatory activities 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 by Favaron et al. in [9]. The atom-bond connectivity \((\mathrm{ABC})\) index, introduced by Estrada et al. [4], gives a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes. The ABC index is used to predict the bioactivity of chemical compounds. The sum-connectivity index was proposed by Zhou et al. [31] and studied in relationship with the Randić index in [1]. Many topological indices have been studied in the literature [327].

For the purpose of this article, we use the following standard notations: Given a graph \(G=(V,E)\) and for \(u,v\in V\), u being adjacent to v is denoted by \(u\sim v\) and \(\{u,v\} \in E\). The distance between any two vertices u and v of G is the minimum number of edges in the shortest \(u-v\) path and is denoted by \(d(u,v)\). The eccentricity \(\operatorname{ecc}(v)\) of a vertex v in G is the maximum distance between v and any of the other vertex of G, i.e. \(\operatorname{ecc}(v)=\max_{u\in V(G)}d(v,u)\). The diameter \(\operatorname{diam}(G)\) of G is the \(\operatorname{diam}(G)=\max_{v\in V(G)}\operatorname{ecc}(v)\) and the radius \(\operatorname{rad}(G)\) of G is the \(\operatorname{rad}(G)=\min_{v\in V(G)}\operatorname{ecc}(v)\). The number of adjacent vertices of v is called the degree of v in G, denoted by \(d_{G}(v)\). The maximum and minimum vertex degree in a graph G, are defined as \(\Delta (G)=\max \{d_{G}(u):u\in V(G)\}\) and \(\delta (G)= \min \{d_{G}(u):u\in V(G)\}\), 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 \(x_{1}x_{2}...x_{t}\) of length t where \(x_{r}\in V\), \(1\leq r\leq t\). \(\{x,y\}\in E(S(G,t))\) if and only if there exists \(i\in \{1,\ldots,n\}\) such that:
  1. 1.

    \(x_{j}=y_{j}\), if \(j< i\);

     
  2. 2.

    \(x_{i}\neq y_{i}\) and \(\{x_{i},y_{i}\}\in E(G)\);

     
  3. 3.

    \(x_{j}=y_{i}\) and \(y_{j}=x_{i}\) if \(j>i\).

     
The \(S(K_{n},3)\) are Tower of Hanoi graphs. The Sierpiński graph is a graph with \(G=K_{n}\), 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 such that \(x=wuvv\cdots v\) and \(y=wvuu\cdots u\). Extreme vertices are the vertices of the form \(uu\cdots u\). Note that, for any graph G of order n and any integer \(t\geq 2\), \(S(G,t)\) has n extreme vertices and, if u has degree \(d_{G}(u)\), then the extreme vertex \(uu\cdots u\) of \(S(G,t)\) also has degree \(d_{G}(u)\). The degrees of vertices of the form \(vuu\cdots u\) and \(uvv\cdots v\) are equal to \(d_{G}(u) + 1\) and \(d_{G}(v) + 1\), respectively. For better understanding of \(S(G,t)\), please see Fig. 1.
Figure 1
Figure 1

Generalized Sierpiński Graphs \(S(1,C_{4})\) and \(S(3,C_{4})\)

It is clear that u as a vertex of \(V(S(G,t))\) has degree \(d_{S(G,t)}(u) \in \{d_{G}(u),d_{G}(u)+1\}\) where \(d_{G}(u)\) is degree of u in G. We use the terminology of [5]. \(\Im _{S(G, t)}(d_{G}(u),d _{G}(v))\) is the number of copies of \(\{u,v\}\) edge with degrees \(d_{G}(u)\) and \(d_{G}(v)\) in \(S(G,t)\). \(N(u)=\{s\in V:\{u,s\}\in E\}\) is the neighborhood of \(u\in V\). For \(u,v\in V\), the number of triangles of G containing u and v will be denoted by \(\tau (u,v)\) and the number of triangles of G will be denoted by \(\tau (G)\). For any pair of adjacent vertices \(u,v\in V\), we have \(|N(u)\cap N(v)|=\tau (u,v)\), \(|N(u)\cup N(v)|=d_{G}(u)+d_{G}(v)-\tau (u,v)\) and \(|N(u)-N(v)|=d_{G}(u)- \tau (u,v)\). From now onward, for a graph of order n, we will use the function \(\psi _{n}(t)=1+n+n^{2}+\cdots +n^{t-1}=\frac{n^{t}-1}{n-1}\).

Lemma 1

([5])

For any integer \(t\geq 2\) and any edge \(\{u,v\}\) of a graph G of order n,
  1. 1.

    \(\Im _{S(G, t)}(d_{G}(u),d_{G}(v))=n^{t-2}(n-d_{G}(u)-d _{G}(v)+\tau (u,v))\).

     
  2. 2.

    \(\Im _{S(G, t)}(d_{G}(u),d_{G}(v)+1)=n^{t-2}(d_{G}(v)- \tau (u,v))-\psi _{n}(t-2)d_{G}(u)\).

     
  3. 3.

    \(\Im _{S(G, t)}(d_{G}(u)+1,d_{G}(v))=n^{t-2}(d_{G}(u)- \tau (u,v))-\psi _{n}(t-2)d_{G}(v)\).

     
  4. 4.

    \(\Im _{S(G, t)}(d_{G}(u)+1,d_{G}(v)+1)=n^{t-2}(\tau (u,v)+1)+ \psi _{n}(t-2)(d_{G}(u)+d_{G}(v)+1)\).

     

Sierpiński graphs \(S(K_{n},t)\) 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 obtained from Sierpiński graphs by adding an open edge to each of its extreme vertices. The graphs \(S(G,t)\) have been studied from different points of views. Colorings of these graphs have been found in [10, 21], crossing number and several metric invariants such as unique 1-perfect codes, average distance of these graphs have been studied in the literature. For more literature on Sierpiński graphs, please see [14, 19, 20, 24, 30].

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

2 Topological indices of generalized Sierpiński graphs

The atom-bond connectivity \((\mathrm{ABC})\) index, introduced by Estrada et al. [4], a degree based index, is defined as
$$ \operatorname{ABC}(G)=\sum_{u\sim v}\sqrt{ \frac{d_{G}(u)+d_{G}(v)-2}{d_{G}(u)d _{G}(v)}}. $$
In the following theorem, we present a lower bound for the atom-bond connectivity index for generalized Sierpiński graphs in terms of the atom-bond connectivity index of G.

Theorem 1

For any graph G of order \(n\geq 3\), \(\delta (G)\geq 2\) and any integer \(t\geq 2\),
$$ \operatorname{ABC}\bigl(S(G,t)\bigr)\leq \psi _{n}(t) \operatorname{ABC}(G). $$

Proof

The ABC index of \(S(G,t)\) can be expressed as
$$ \operatorname{ABC}\bigl(S(G,t)\bigr)=\sum_{{u\sim v}}\sum _{i=0}^{1}\sum_{j=0}^{1} \Im _{S(G,t)}\bigl(d_{G}(u)+i,d_{G}(v)+j\bigr)\sqrt{ \frac{(d_{G}(u)+i)+(d _{G}(v)+j)-2}{(d_{G}(u)+i)(d_{G}(v)+j)}} $$
by using Lemma 1:
$$ \begin{aligned} &=\sum_{u\sim v}\biggl\{ n^{t-2} \bigl(n-d_{G}(u)-d_{G}(v)+\tau (u,v)\bigr)\sqrt{ \frac{d _{G}(u)+d_{G}(v)-2}{d_{G}(u)d_{G}(v)}} \\ &\quad {}+n^{t-2}\bigl(d_{G}(v)-\tau (u,v)\bigr)- \psi _{n}(t-2)d_{G}(u)\sqrt{\frac{d_{G}(u)+d_{G}(v)-1}{d_{G}(u)(d _{G}(v)+1)}} \\ &\quad {}+n^{t-2}\bigl(d_{G}(u)-\tau (u,v)\bigr)-\psi _{n}(t-2)d_{G}(v)\sqrt{\frac{d _{G}(u)+d_{G}(v)-1}{(d_{G}(u)+1)d_{G}(v)}} \\ &\quad {}+n^{t-2}\bigl(\tau (u,v)+1\bigr)+ \psi _{n}(t-2) \bigl(d_{G}(u)+d_{G}(v)+1\bigr)\sqrt{\frac{d_{G}(u)+d_{G}(v)}{(d _{G}(u)+1)(d_{G}(v)+1)}} \biggr\} . \end{aligned} $$
We have \(\delta (G)\geq 2\), \(\sqrt{\frac{d_{G}(u)+d_{G}(v)-1}{d_{G}(u)(d _{G}(v)+1)}}\leq \sqrt{\frac{d_{G}(u)+d_{G}(v)-2}{d_{G}(u)d_{G}(v)}}\), \(\sqrt{\frac{d _{G}(u)+d_{G}(v)-1}{(d_{G}(u)+1)d_{G}(v)}}\leq \sqrt{\frac{d_{G}(u)+d _{G}(v)-2}{d_{G}(u)d_{G}(v)}}\) and \(\sqrt{\frac{d_{G}(u)+d_{G}(v)}{(d _{G}(u)+1)(d_{G}(v)+1)}}\leq \sqrt{\frac{d_{G}(u)+d_{G}(v)-2}{d_{G}(u)d _{G}(v)}}\). We deduce
$$ \begin{aligned} &\operatorname{ABC}\bigl(S(G,t)\bigr)\\ &\quad \leq \sum _{u\sim v}\bigl\{ n^{t-2}\bigl(n-d_{G}(u)-d_{G}(v)+ \tau (u,v)\bigr) +n^{t-2}\bigl(d_{G}(v)-\tau (u,v)\bigr)\\ &\qquad{}-\psi _{n}(t-2)d_{G}(u)+n^{t-2}\bigl(d _{G}(u)-\tau (u,v)\bigr)-\psi _{n}(t-2)d_{G}(v) \\ &\qquad {}+n^{t-2}\bigl(\tau (u,v)+1\bigr)+\psi _{n}(t-2) \bigl(d_{G}(u)+d_{G}(v)+1\bigr)\bigr\} \sqrt{ \frac{d _{G}(u)+d_{G}(v)-2}{d_{G}(u)d_{G}(v)}} \\ &\quad =\sum_{u\sim v}\bigl\{ n^{t-1}+n^{t-2}+ \psi _{n}(t-2)\bigr\} \sqrt{\frac{d _{G}(u)+d_{G}(v)-2}{d_{G}(u)d_{G}(v)}} \\ &\quad =\sum_{u\sim v}\psi _{n}(t)\sqrt{ \frac{d_{G}(u)+d_{G}(v)-2}{d _{G}(u)d_{G}(v)}}, \end{aligned} $$
which gives the desired result:
$$ \operatorname{ABC}\bigl(S(G,t)\bigr)\leq \psi _{n}(t) \operatorname{ABC}(G). $$
 □
The harmonic index, another variant of degree based topological indices, is defined as follows:
$$ H(G)=\sum_{u\sim v}\frac{2}{d_{G}(u)+d_{G}(v)}. $$
Now, we compute a lower bound for the harmonic index for generalized Sierpiński graphs.

Theorem 2

For any graph G of order \(n\geq 2\), size \(m\geq 1\) and any integer \(t\geq 2\),
$$ H\bigl(S(G,t)\bigr)\leq \psi _{n}(t)H(G). $$

Proof

The \(H(G)\) index of \(S(G,t)\) can be expressed as
$$ H\bigl(S(G,t)\bigr)=\sum_{u\sim v}\sum _{i=0}^{1}\sum_{j=0} ^{1}\Im _{S(G,t)}\bigl(d_{G}(u)+i,d_{G}(v)+j \bigr) \frac{2}{(d_{G}(u)+i)+(d_{G}(v)+j)}. $$
By using Lemma 1 and \(\frac{2}{d_{G}(u)+d_{G}(v)+j}\leq \frac{2}{d _{G}(u)+d_{G}(v)}\), for \(j=1,2\). We have
$$ \begin{aligned} H\bigl(S(G,t)\bigr)&\leq \sum_{u\sim v} \bigl\{ n^{t-2}\bigl(n-d_{G}(u)-d_{G}(v)+ \tau (u,v) \bigr) +n^{t-2}\bigl(d_{G}(v)-\tau (u,v)\bigr)\\&\quad{}-\psi _{n}(t-2)d_{G}(u)+n^{t-2}\bigl(d _{G}(u)-\tau (u,v)\bigr)-\psi _{n}(t-2)d_{G}(v) \\ &\quad {}+n^{t-2}\bigl(\tau (u,v)+1\bigr)+\psi _{n}(t-2) \bigl(d_{G}(u)+d_{G}(v)+1\bigr)\bigr\} \frac{2}{d _{G}(u)+d_{G}(v)} \\ &=\sum_{u\sim v}\bigl\{ n^{t-1}+n^{t-2}+ \psi _{n}(t-2)\bigr\} \frac{2}{d _{G}(u)+d_{G}(v)} \\ &=\sum_{u\sim v}\psi _{n}(t) \frac{2}{d_{G}(u)+d_{G}(v)}. \end{aligned} $$
We get the required result:
$$ H\bigl(S(G,t)\bigr)\leq \psi _{n}(t)H(G). $$
 □
Bounds on the Zagreb first and second indices, defined as
$$ Z_{1}(G)=\sum_{u\sim v}\bigl(d_{G}(u)+d_{G}(v) \bigr) $$
and
$$ Z_{2}(G)=\sum_{u\sim v}\bigl(d_{G}(u) \cdot d_{G}(v)\bigr) $$
for generalized Sierpiński graphs are presented in the following theorems.

Theorem 3

For any graph G of order \(n\geq 2\), size \(m\geq 1\) and any integer \(t\geq 2\),
$$ Z_{1}\bigl(S(G,t)\bigr)\leq 2m\psi _{n}(t) \bigl(n- \operatorname{rad}(G)+1\bigr). $$

Proof

The \(Z_{1}(G)\) index of \(S(G,t)\) can be expressed as
$$ Z_{1}\bigl(S(G,t)\bigr)=\sum_{u\sim v}\sum _{i=0}^{1}\sum_{j=0}^{1} \Im _{S(G,t)}\bigl(d_{G}(u)+i,d_{G}(v)+j\bigr) \bigl( \bigl(d_{G}(u)+i\bigr)+\bigl(d_{G}(v)+j\bigr)\bigr). $$
It is clear that \(d_{G}(u)+d_{G}(v)\leq d_{G}(u)+d_{G}(v)+1\leq d_{G}(u)+d _{G}(v)+2\). Using Lemma 1 and the above inequality we get
$$ \begin{aligned} Z_{1}\bigl(S(G,t)\bigr)&\leq \sum _{u\sim v}\bigl\{ n^{t-2}\bigl(n-d_{G}(u)-d_{G}(v)+ \tau (u,v)\bigr) \\ &\quad {}+n^{t-2}\bigl(d_{G}(v)-\tau (u,v)\bigr)-\psi _{n}(t-2)d_{G}(u) +n^{t-2}\bigl(d_{G}(u)- \tau (u,v)\bigr)\\&\quad{}-\psi _{n}(t-2)d_{G}(v)+n^{t-2} \bigl(\tau (u,v)+1\bigr)\\&\quad{}+\psi _{n}(t-2) \bigl(d _{G}(u)+d_{G}(v)+1 \bigr)\bigr\} \bigl(d_{G}(u)+d_{G}(v)+2\bigr) \\ &=\sum_{u\sim v}\bigl\{ n^{t-1}+n^{t-2}+ \psi _{n}(t-2)\bigr\} \bigl(d_{G}(u)+d _{G}(v)+2 \bigr) \\ &=\sum_{u\sim v}\psi _{n}(t) \bigl(d_{G}(u)+d_{G}(v)+2\bigr). \end{aligned} $$
By using \(d_{G}(u)\leq n-\operatorname{ecc}(u)\) and \(\operatorname{rad}(G)\leq \operatorname{ecc}(u)\), we get
$$ d_{G}(u)+d_{G}(v)+2\leq 2\bigl(n-\operatorname{rad}(G)+1 \bigr). $$
Using the above inequality, we get the required result:
$$ Z_{1}\bigl(S(G,t)\bigr)\leq 2m\psi _{n}(t) \bigl(n- \operatorname{rad}(G)+1\bigr). $$
 □

Theorem 4

For any graph G of order \(n\geq 2\), size \(m\geq 1\) and any integer \(t\geq 2\),
$$ Z_{2}\bigl(S(G,t)\bigr)\leq m\psi _{n}(t) \bigl(n- \operatorname{rad}(G)+1\bigr)^{2}. $$

Proof

The \(Z_{2}(G)\) index of \(S(G,t)\) can be expressed as
$$ Z_{2}\bigl(S(G,t)\bigr)=\sum_{u\sim v}\sum _{i=0}^{1}\sum_{j=0}^{1} \Im _{S(G,t)}\bigl(d_{G}(u)+i,d_{G}(v)+j\bigr) \bigl( \bigl(d_{G}(u)+i\bigr)\cdot \bigl(d _{G}(v)+j\bigr)\bigr). $$
It is clear that \(d_{G}(u)\cdot d_{G}(v)\leq d_{G}(u)\cdot (d_{G}(v)+1) \leq (d_{G}(u)+1)\cdot d_{G}(v)\leq (d_{G}(u)+1)\cdot {(d_{G}(v)+1)}\). Using Lemma 1 and the above inequality we get
$$\begin{aligned} &Z_{2}\bigl(S(G,t)\bigr)\\ &\quad \leq \sum _{u\sim v}\bigl\{ n^{t-2}\bigl(n-d_{G}(u)-d_{G}(v)+ \tau (u,v)\bigr) +n^{t-2}\bigl(d_{G}(v)-\tau (u,v)\bigr)\\ &\qquad{}-\psi _{n}(t-2)d_{G}(u)+n^{t-2}\bigl(d _{G}(u)-\tau (u,v)\bigr)-\psi _{n}(t-2)d_{G}(v) \\ &\qquad {}+n^{t-2}\bigl(\tau (u,v)+1\bigr)+\psi _{n}(t-2) \bigl(d_{G}(u)+d_{G}(v)+1\bigr)\bigr\} \bigl(d_{G}(u)+1 \bigr) \cdot \bigl(d_{G}(v)+1\bigr) \\ &\quad =\sum_{u\sim v}\bigl\{ n^{t-1}+n^{t-2}+ \psi _{n}(t-2)\bigr\} \bigl(d_{G}(u)+1\bigr) \cdot \bigl(d_{G}(v)+1\bigr) \\ &\quad =\sum_{u\sim v}\psi _{n}(t) \bigl(d_{G}(u)+1\bigr)\cdot \bigl(d_{G}(v)+1\bigr). \end{aligned}$$
We have \((d_{G}(u)+1)\cdot (d_{G}(v)+1)\leq (n-\operatorname{rad}(G)+1)^{2}\). Using the above inequality, we get the required result:
$$ Z_{2}\bigl(S(G,t)\bigr)\leq m\psi _{n}(t) \bigl(n- \operatorname{rad}(G)+1\bigr)^{2}. $$
 □
The sum-connectivity index was proposed by Zhou et al. [31]. The sum-connectivity index \(\chi (G)\) is defined as
$$ \chi (G)=\sum_{u\sim v}\frac{1}{\sqrt{d_{G}(u)+d_{G}(v)}}. $$
In the following result, we discuss the sum-connectivity index of generalized Sierpiński graphs.

Theorem 5

For any graph G of order \(n\geq 2\), size \(m\geq 1\) and any integer \(t\geq 2\),
$$ \chi \bigl(S(G,t)\bigr)\leq \psi _{n}(t)\chi (G). $$

Proof

The \(\chi (S(G,t))\) index of \(S(G,t)\) can be expressed as
$$ \chi \bigl(S(G,t)\bigr)=\sum_{u\sim v}\sum _{i=0}^{1}\sum_{j=0}^{1} \Im _{S(G,t)}\bigl(d_{G}(u)+i,d_{G}(v)+j\bigr) \frac{1}{(d_{G}(u)+i)+(d _{G}(v)+j)}. $$
We have \(\frac{1}{\sqrt{d_{G}(u)+d_{G}(v)+1}}\leq \frac{1}{\sqrt{d _{G}(u)+d_{G}(v)}}\) and \(\frac{1}{\sqrt{d_{G}(u)+d_{G}(v)+2}}\leq \frac{1}{\sqrt{d _{G}(u)+d_{G}(v)}}\). By using Lemma 1 and these inequalities, we have
$$ \begin{aligned} \chi \bigl(S(G,t)\bigr)&\leq \sum_{u\sim v} \bigl\{ n^{t-2}\bigl(n-d_{G}(u)-d_{G}(v)+ \tau (u,v) \bigr) +n^{t-2}\bigl(d_{G}(v)-\tau (u,v)\bigr)\\&\quad{}-\psi _{n}(t-2)d_{G}(u)+n^{t-2}\bigl(d _{G}(u)-\tau (u,v)\bigr)-\psi _{n}(t-2)d_{G}(v) \\ &\quad {}+n^{t-2}\bigl(\tau (u,v)+1\bigr)+\psi _{n}(t-2) \bigl(d_{G}(u)+d_{G}(v)+1\bigr)\bigr\} \frac{1}{\sqrt{d _{G}(u)+d_{G}(v)}} \\ &=\sum_{u\sim v}\bigl\{ n^{t-1}+n^{t-2}+ \psi _{n}(t-2)\bigr\} \frac{1}{\sqrt{d _{G}(u)+d_{G}(v)}} \\ &=\sum_{u\sim v}\psi _{n}(t) \frac{1}{\sqrt{d_{G}(u)+d_{G}(v)}}, \end{aligned} $$
which gives the desired result:
$$ \chi \bigl(S(G,t)\bigr)\leq \psi _{n}(t)\chi (G). $$
 □

3 Topological indices of extended Sierpiński graphs

The extended Sierpiński graph \(S(G,t^{+})\) was introduced by Klavžar et al. in [25]. The extended Sierpiński graph \(S(G,t^{+})\) is obtained from \(S(G,t)\) by adding a new vertex x, called the special vertex of \(S(G,t^{+})\), and edges joining x with all extreme vertices ϖ of \(S(G,t)\), as shown in Fig. 2. These edges are \(d_{S(G,t)}(s)\in \{d_{G}(s),d_{G}(s)+1\}\). The number of extreme vertices in \(S(G,t)\) is always the order of G. Therefore, in analogy to Lemma 1, we give the following result for extended Sierpiński graphs.
Figure 2
Figure 2

Extended Sierpiński graph

Lemma 2

For any integer \(t\geq 2\) and any edge \(\{u,v\}\) of a graph G of order n.
  1. 1.

    \(\Im _{S(G,t^{+})}(d_{G}(u),d_{G}(v))=n^{t-2}(n-d_{G}(u)-d _{G}(v)+\tau (u,v))\).

     
  2. 2.

    \(\Im _{S(G,t^{+})}(d_{G}(u),d_{G}(v)+1)=n^{t-2}(d_{G}(v)- \tau (u,v))-\psi _{n}(t-2)d_{G}(u)\).

     
  3. 3.

    \(\Im _{S(G,t^{+})}(d_{G}(u)+1,d_{G}(v))=n^{t-2}(d_{G}(u)- \tau (u,v))-\psi _{n}(t-2)d_{G}(v)\).

     
  4. 4.

    \(\Im _{S(G,t^{+})}(d_{G}(u)+1,d_{G}(v))=n^{t-2}(\tau (u,v)+1)+ \psi _{n}(t-2)(d_{G}(u)+d_{G}(v)+1)\).

     
  5. 5.

    \(\Im _{S(G,t^{+})}(d_{G}(\varpi ),d_{S(G,t^{+})}(x))=n\).

     

We use the above lemma to find topological indices for extended Sierpiński graphs \(S(G,t^{+})\).

Theorem 6

For any graph G of order \(n\geq 3\), \(\delta (G)\geq 2\) and any integer \(t\geq 2\),
$$ \operatorname{ABC}\bigl(S\bigl(G,t^{+}\bigr)\bigr)\leq \bigl(n +\psi _{n}(t)\bigr)\operatorname{ABC}(G). $$

Proof

The ABC index of \(S(G,t^{+})\) can be expressed as
$$ \begin{aligned} &\operatorname{ABC}\bigl(S\bigl(G,t^{+}\bigr)\bigr)\\ &\quad = \sum_{u\sim v}\sum_{i=0}^{1} \sum_{j=0}^{1}\Im _{S(G,t^{+})} \bigl(d_{G}(u)+i,d_{G}(v)+j\bigr)\sqrt{\frac{(d _{G}(u)+i)+(d_{G}(v)+j)-2}{(d_{G}(u)+i)(d_{G}(v)+j)}} \\ &\qquad {}+\Im _{S(G,t^{+})}\bigl(d_{G}(\varpi ),d_{S(G,t^{+})}(x) \bigr)\sqrt{\frac{d _{G}(\varpi )+d_{S(G,t^{+})}(x)-2}{d_{G}(\varpi )d_{S(G,t^{+})}(x)}}; \end{aligned} $$
by using Lemma 2
$$ \begin{aligned} &\operatorname{ABC}\bigl(S\bigl(G,t^{+}\bigr)\bigr)\\ &\quad = \sum_{u\sim v}\biggl\{ n^{t-2}\bigl(n-d_{G}(u)-d_{G}(v)+ \tau (u,v)\bigr)\sqrt{\frac{d_{G}(u)+d_{G}(v)-2}{d_{G}(u)d_{G}(v)}} \\ &\qquad {}+n^{t-2}\bigl(d_{G}(v)-\tau (u,v)\bigr)-\psi _{n}(t-2)d_{G}(u)\sqrt{\frac{d _{G}(u)+d_{G}(v)-1}{d_{G}(u)(d_{G}(v)+1)}} \\ &\qquad{}+n^{t-2}\bigl(d_{G}(u)-\tau (u,v)\bigr)- \psi _{n}(t-2)d_{G}(v)\sqrt{\frac{d_{G}(u)+d_{G}(v)-1}{(d_{G}(u)+1)d _{G}(v)}} \\ &\qquad {}+n^{t-2}\bigl(\tau (u,v)+1\bigr)+\psi _{n}(t-2) \bigl(d_{G}(u)+d_{G}(v)+1\bigr)\sqrt{\frac{d _{G}(u)+d_{G}(v)}{(d_{G}(u)+1)(d_{G}(v)+1)}} \\ &\qquad {}+n\sqrt{\frac{d_{G}( \varpi )+d_{S(G,t^{+})}(x)-2}{d_{G}(\varpi )d_{S(G,t^{+})}(x)}}\biggr\} . \end{aligned} $$
We have \(\delta (G)\geq 2\), \(\sqrt{\frac{d_{G}(u)+d_{G}(v)-1}{d_{G}(u)(d _{G}(v)+1)}}\leq \sqrt{\frac{d_{G}(u)+d_{G}(v)-2}{d_{G}(u)d_{G}(v)}}\), \(\sqrt{\frac{d _{G}(u)+d_{G}(v)-1}{(d_{G}(u)+1)d_{G}(v)}}\leq \sqrt{\frac{d_{G}(u)+d _{G}(v)-2}{d_{G}(u)d_{G}(v)}}\), \(\sqrt{\frac{d_{G}(u)+d_{G}(v)}{(d _{G}(u)+1)(d_{G}(v)+1)}}\leq \sqrt{\frac{d_{G}(u)+d_{G}(v)-2}{d_{G}(u)d _{G}(v)}}\) and \(\sqrt{\frac{d_{G}(\varpi )+d_{S(G,t^{+})}(x)-2}{d _{G}(\varpi )d_{S(G,t^{+})}(x)}}\leq \sqrt{\frac{d_{G}(u)+d_{G}(v)-2}{d _{G}(u)d_{G}(v)}}\). We have
$$ \begin{aligned} &\operatorname{ABC}\bigl(S\bigl(G,t^{+}\bigr)\bigr)\\ &\quad \leq \sum_{u\sim v}\bigl\{ n^{t-2} \bigl(n-d_{G}(u)-d_{G}(v)+ \tau (u,v)\bigr) +n^{t-2} \bigl(d_{G}(v)-\tau (u,v)\bigr)\\ &\qquad {}-\psi _{n}(t-2)d_{G}(u)+n^{t-2} \bigl(d _{G}(u)-\tau (u,v)\bigr)-\psi _{n}(t-2)d_{G}(v) \\ &\qquad {}+n^{t-2} \bigl(\tau (u,v)+1\bigr)+\psi _{n}(t-2) \bigl(d_{G}(u)+d_{G}(v)+1\bigr)+n\bigr\} \sqrt{ \frac{d _{G}(u)+d_{G}(v)-2}{d_{G}(u)d_{G}(v)}} \\ &\quad =\sum_{u\sim v}\bigl\{ n^{t-1}+n^{t-2}+ \psi _{n}(t-2)+n\bigr\} \sqrt{\frac{d _{G}(u)+d_{G}(v)-2}{d_{G}(u)d_{G}(v)}} \\ &\quad =\sum_{u\sim v}\bigl(n+\psi _{n}(t)\bigr) \sqrt{\frac{d_{G}(u)+d_{G}(v)-2}{d _{G}(u)d_{G}(v)}} \end{aligned} $$
and we get the required result:
$$ \operatorname{ABC}\bigl(S\bigl(G,t^{+}\bigr)\bigr)\leq \bigl(n +\psi _{n}(t)\bigr)\operatorname{ABC}(G). $$
 □

A lower bound for the harmonic index of extended Sierpiński graphs \(S(G,t^{+})\) is given in the following result.

Theorem 7

For any graph G of order \(n\geq 2\), size \(m\geq 1\) and any integer \(t\geq 2\),
$$ H\bigl(S\bigl(G,t^{+}\bigr)\bigr)\leq \bigl(n+\psi _{n}(t) \bigr)H(G). $$

Proof

The \(H(G)\) index of \(S(G,t^{+})\) can be expressed as
$$ \begin{aligned} H\bigl(S\bigl(G,t^{+}\bigr)\bigr)&=\sum _{u\sim v}\sum_{i=0}^{1}\sum _{j=0}^{1}\Im _{S(G,t^{+})} \bigl(d_{G}(u)+i,d_{G}(v)+j\bigr)\frac{2}{(d_{G}(u)+i)+(d _{G}(v)+j)} \\ &\quad {}+\Im _{S(G,t^{+})}\bigl(d_{G}(\varpi ),d_{S(G,t^{+})}(x) \bigr)\frac{2}{d _{G}(\varpi )+d_{S(G,t^{+})}(x)}. \end{aligned} $$
We have \(\frac{2}{d_{G}(u)+d_{G}(v)+j}\leq \frac{2}{d_{G}(u)+d_{G}(v)}\), for \(j=1,2\) and \(\frac{2}{d_{G}(\varpi )+d_{S(G,t^{+})}(x)}\leq \frac{2}{d_{G}(u)+d_{G}(v)}\). By using Lemma 2 and the above inequalities, we have
$$ \begin{aligned} H\bigl(S\bigl(G,t^{+}\bigr)\bigr)&\leq \sum _{u\sim v}\bigl\{ n^{t-2}\bigl(n-d_{G}(u)-d_{G}(v)+ \tau (u,v)\bigr) +n^{t-2}\bigl(d_{G}(v)-\tau (u,v)\bigr)\\&\quad{}-\psi _{n}(t-2)d_{G}(u)+n^{t-2}\bigl(d _{G}(u)-\tau (u,v)\bigr)-\psi _{n}(t-2)d_{G}(v) \\ &\quad {}+n^{t-2} \bigl(\tau (u,v)+1\bigr)+\psi _{n}(t-2) \bigl(d_{G}(u)+d_{G}(v)+1\bigr)+n\bigr\} \frac{2}{d _{G}(u)+d_{G}(v)} \\ &=\sum_{u\sim v}\bigr\{ n^{t-1}+n^{t-2}+\psi _{n}(t-2)+ n\bigr\} \frac{2}{d_{G}(u)+d_{G}(v)} \\ &=\sum_{u\sim v}\bigl(n+\psi _{n}(t)\bigr) \frac{2}{d_{G}(u)+d_{G}(v)}, \end{aligned} $$
which gives the desired result:
$$ H\bigl(S\bigl(G,t^{+}\bigr)\bigr)\leq \bigl(n+\psi _{n}(t) \bigr)H(G). $$
 □

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

Theorem 8

For any graph G of order \(n\geq 2\), size \(m\geq 1\) and any integer \(t\geq 2\),
$$ Z_{1}\bigl(S\bigl(G,t^{+}\bigr)\bigr)\leq 2m\bigl(n+\psi _{n}(t)\bigr) \bigl(n-\operatorname{rad}(G)+1\bigr). $$

Proof

The \(Z_{1}\) index of \(S(G,t^{+})\) can be expressed as
$$ \begin{aligned} Z_{1}\bigl(S\bigl(G,t^{+}\bigr)\bigr)&=\sum _{u\sim v}\sum_{i=0}^{1} \sum_{j=0}^{1}\Im _{S(G,t)} \bigl(d_{G}(u)+i,d_{G}(v)+j\bigr) \bigl( \bigl(d_{G}(u)+i\bigr)+\bigl(d _{G}(v)+j\bigr)\bigr) \\ &\quad {}+\Im _{S(G,t^{+})}\bigl(d_{G}(\varpi ),d_{S(G,t^{+})}(x) \bigr) \bigl(d _{G}(\varpi )+d_{S(G,t^{+})}(x)\bigr). \end{aligned} $$
It is clear that \(d_{G}(u)+d_{G}(v)\leq d_{G}(u)+d_{G}(v)+1\leq d_{G}(u)+d _{G}(v)+2\) and similarly \((d_{G}(\varpi )+d_{S(G,t^{+})}(x))\leq d _{G}(u)+d_{G}(v)+2\). Using Lemma 2 and the above inequalities we get
$$ \begin{aligned} &Z_{1}\bigl(S\bigl(G,t^{+}\bigr)\bigr)\\ &\quad \leq \sum_{u\sim v}\bigl\{ n^{t-2}\bigl(n-d_{G}(u)-d _{G}(v)+\tau (u,v)\bigr) +n^{t-2}\bigl(d_{G}(v)- \tau (u,v)\bigr)\\ &\qquad{}-\psi _{n}(t-2)d_{G}(u)+n ^{t-2} \bigl(d_{G}(u)-\tau (u,v)\bigr)- \psi _{n}(t-2)d_{G}(v) \\ &\qquad {}+n^{t-2}\bigl(\tau (u,v)+1\bigr)+\psi _{n}(t-2) \bigl(d_{G}(u)+d_{G}(v)+1\bigr)+n\bigr\} \bigl(d_{G}(u)+d _{G}(v)+2\bigr) \\ &\quad =\sum_{u\sim v}\bigl\{ n^{t-1}+n^{t-2}+ \psi _{n}(t-2)+n\bigr\} \bigl(d_{G}(u)+d _{G}(v)+2 \bigr) \\ &\quad =\sum_{u\sim v}\bigl(n+\psi _{n}(t)\bigr) \bigl(d_{G}(u)+d_{G}(v)+2\bigr). \end{aligned} $$
We have \(d_{G}(u)+d_{G}(v)+2\leq 2(n-\operatorname{rad}(G)+1)\) Using the above inequality, we get the required result:
$$ Z_{1}\bigl(S\bigl(G,t^{+}\bigr)\bigr)\leq 2m\bigl(n+\psi _{n}(t)\bigr) \bigl(n-\operatorname{rad}(G)+1\bigr). $$
 □

Theorem 9

For any graph G of order \(n\geq 2\), size \(m\geq 1\) and any integer \(t\geq 2\),
$$ Z_{2}\bigl(S\bigl(G,t^{+}\bigr)\bigr)\leq m\bigl(n+\psi _{n}(t)\bigr) \bigl(n-\operatorname{rad}(G)+1\bigr)^{2}. $$

Proof

The \(Z_{2}\) index of \(S(G,t^{+})\) can be expressed as
$$ \begin{aligned} Z_{2}\bigl(S\bigl(G,t^{+}\bigr)\bigr)&=\sum _{u\sim v}\sum_{i=0}^{1} \sum_{j=0}^{1}\Im _{S(G,t^{+})} \bigl(d_{G}(u)+i,d_{G}(v)+j\bigr) \bigl( \bigl(d_{G}(u)+i\bigr) \cdot \bigl(d_{G}(v)+j\bigr)\bigr) \\ &\quad {}+\Im _{S(G,t^{+})}\bigl(d_{G}(\varpi ),d_{S(G,t^{+})}(x) \bigr) \bigl(d _{G}(\varpi )\bigr)\cdot \bigl(d_{S(G,t^{+})}(x) \bigr). \end{aligned} $$
It is clear that \(d_{G}(u)\cdot d_{G}(v)\leq d_{G}(u)\cdot (d_{G}(v)+1) \leq (d_{G}(u)+1)\cdot d_{G}(v)\leq (d_{G}(u)+1)\cdot (d_{G}(v)+1)\) and \(((d_{G}(\varpi ))\cdot (d_{S(G,t^{+})}(x)))\leq (d_{G}(u)+1)\cdot (d _{G}(v)+1)\). Using Lemma 2 and the above inequalities, we get
$$ \begin{aligned} Z_{2}\bigl(S\bigl(G,t^{+}\bigr)\bigr)&\leq \sum_{u\sim v}\bigl\{ n^{t-2}\bigl(n-d_{G}(u)-d _{G}(v)+\tau (u,v)\bigr) \\ &\quad {}+n^{t-2}\bigl(d_{G}(v)-\tau (u,v)\bigr)-\psi _{n}(t-2)d_{G}(u) +n^{t-2}\bigl(d_{G}(u)- \tau (u,v)\bigr)\\&\quad{}-\psi _{n}(t-2)d_{G}(v)+n^{t-2} \bigl(\tau (u,v)+1\bigr)\\&\quad{}+\psi _{n}(t-2) \bigl(d _{G}(u)+d_{G}(v)+1 \bigr)+n\bigr\} \bigl(d_{G}(u)+1\bigr)\cdot \bigl(d_{G}(v)+1 \bigr) \\ &=\sum_{u\sim v}\bigl\{ n^{t-1}+n^{t-2}+ \psi _{n}(t-2)+n\bigr\} \bigl(d_{G}(u)+1\bigr) \cdot \bigl(d_{G}(v)+1\bigr) \\ &=\sum_{u\sim v}\bigl(n+\psi _{n}(t)\bigr) \bigl(d_{G}(u)+1\bigr)\cdot \bigl(d_{G}(v)+1\bigr). \end{aligned} $$
We have \((d_{G}(u)+1)\cdot (d_{G}(v)+1)\leq (n-\operatorname{rad}(G)+1)^{2}\). Using this inequality, we get the required result:
$$ Z_{2}\bigl(S\bigl(G,t^{+}\bigr)\bigr)\leq m\bigl(n+\psi _{n}(t)\bigr) \bigl(n-\operatorname{rad}(G)+1\bigr)^{2}. $$
 □

The sum-connectivity index of extended Sierpiński graphs \(S(G,t^{+})\) is calculated in the following result.

Theorem 10

For any graph G of order \(n\geq 2\), size \(m\geq 1\) and any integer \(t\geq 2\),
$$ \chi \bigl(S\bigl(G,t^{+}\bigr)\bigr)\leq \bigl(n+\psi _{n}(t)\bigr)\chi (G). $$

Proof

The \(\chi (S(G,t^{+}))\) index of \(S(G,t^{+})\) can be expressed as
$$ \begin{aligned} \chi \bigl(S\bigl(G,t^{+}\bigr)\bigr)&=\sum _{u \sim v}\sum_{i=0}^{1}\sum _{j=0}^{1}\Im _{S(G,t)} \bigl(d_{G}(u)+i,d_{G}(v)+j\bigr)\frac{1}{\sqrt{(d _{G}(u)+i)+(d_{G}(v)+j)}} \\ &\quad {}+\Im _{S(G,t)}\bigl(d_{G}(\varpi ),d_{S(G,t^{+})}(x) \bigr)\frac{1}{\sqrt{d _{G}(\varpi )+d_{S(G,t^{+})}(x)}}. \end{aligned} $$
We have \(\frac{1}{\sqrt{d_{G}(u)+d_{G}(v)+1}}\leq \frac{1}{\sqrt{d _{G}(u)+d_{G}(v)}}\), \(\frac{1}{\sqrt{d_{G}(u)+d_{G}(v)+2}}\leq \frac{1}{\sqrt{d _{G}(u)+d_{G}(v)}}\) and \(\frac{1}{\sqrt{d_{G}(\varpi )+d_{S(G,t^{+})}(x)}}\leq \frac{1}{\sqrt{d _{G}(u)+d_{G}(v)}}\). By using Lemma 2 and these inequalities we have
$$ \begin{aligned} \chi \bigl(S\bigl(G,t^{+}\bigr)\bigr)&\leq \sum _{u\sim v}\bigl\{ n^{t-2}\bigl(n-d_{G}(u)-d _{G}(v)+\tau (u,v)\bigr) +n^{t-2}\bigl(d_{G}(v)- \tau (u,v)\bigr) \\ &\quad {}-\psi _{n}(t-2)d_{G}(u)+n ^{t-2} \bigl(d_{G}(u)-\tau (u,v)\bigr)- \psi _{n}(t-2)d_{G}(v) \\ &\quad {}+n^{t-2}\bigl(\tau (u,v)+1\bigr)+\psi _{n}(t-2) \bigl(d_{G}(u)+d_{G}(v)+1\bigr)+n\bigr\} \frac{1}{\sqrt{d _{G}(u)+d_{G}(v)}} \\ &=\sum_{u\sim v}\bigl\{ n^{t-1}+n^{t-2}+ \psi _{n}(t-2)+n\bigr\} \frac{1}{\sqrt{d _{G}(u)+d_{G}(v)}} \\ &=\sum_{u\sim v}\bigl(n+\psi _{n}(t)\bigr) \frac{1}{ \sqrt{d_{G}(u)+d_{G}(v)}}, \end{aligned} $$
which gives the desired result:
$$ \chi \bigl(S\bigl(G,t^{+}\bigr)\bigr)\leq \bigl(n+\psi _{n}(t)\bigr)\chi (G). $$
 □

4 Conclusion

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

Declarations

Acknowledgements

Not applicable.

Availability of data and materials

Not applicable.

Funding

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

Authors’ contributions

All authors contributed equally and significantly in conducting this research work and writing this paper. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access 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 any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan
(2)
Department of Mathematical Sciences, United Arab Emirates University, Al Ain, United Arab Emirates
(3)
Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad, Pakistan
(4)
Department of Mathematics, University of Education Lahore Campus Dera Ghazi Khan, Dera Ghazi Khan, Pakistan

References

  1. Das, K.C., Das, S., Zhou, B.: Sum-connectivity index of a graph. Front. Math. China 11(1), 47–54 (2016) MathSciNetMATHView ArticleGoogle Scholar
  2. Della Vecchia, G., Sanges, C.: A recursively scalable network VLSI implementation. Future Gener. Comput. Syst. 4, 235–243 (1988) View ArticleGoogle Scholar
  3. Eliasi, M., Iranmanesh, A., Gutman, I.: Multiplicative versions of first Zagreb index. MATCH Commun. Math. Comput. Chem. 68, 217–230 (2012) MathSciNetMATHGoogle Scholar
  4. Estrada, E., Torres, L., Rodríguez, L., Gutman, I.: An atom bond connectivity: modelling the entholpy of formation of alkanes, I. Indian J. Chem. 37A, 849 (1998) Google Scholar
  5. Estrada-Moreno, A., Rodríguez-Velázquez, J.A.: On the general Randić index of polymeric networks modelled by generalized Sierpiński graphs. Discrete Appl. Math. (2018) Google Scholar
  6. Farahani, M.R.: Some connectivity indices and Zagreb indices of polyhex nanotubes. Acta Chim. Slov. 59, 779–783 (2012) MathSciNetGoogle Scholar
  7. Farahani, M.R.: On multiple Zagreb indices of circumcoronene homologous series of benzenoid. J. Chem. Phys. 7(2), 277–281 (2014) Google Scholar
  8. Farahani, M.R.: Zagreb indices and their polynomials of the linear parallelogram of benzenoid graph. Global J. Chem. 1(1), 16–19 (2015) Google Scholar
  9. Favaron, O., Mahéo, M., Saclé, J.F.: Some eigenvalue properties in graphs (conjectures of Graffiti 11). Discrete Math. 111, 197–220 (1993) MathSciNetMATHView ArticleGoogle Scholar
  10. Fu, H.-Y.: P r ${P_{r}}$ -free colorings of Sierpiński-like graphs. Ars Comb. 513–524 (2012) Google Scholar
  11. Fu, J.-S.: Hamiltonian connectivity of the WK-recursive network with faulty nodes. Inf. Sci. 178 2573–2584 (2008) MathSciNetMATHView ArticleGoogle Scholar
  12. Ghorbani, M., Azimi, N.: Note on multiple Zagreb indices. Iran. J. Math. Chem. 3, 137–143 (2012) MATHGoogle Scholar
  13. Godsil, C., Royle, G.: Algebraic Graph Theory. Springer, New York (2001) MATHView ArticleGoogle Scholar
  14. Gravier, S., Kovše, M., Mollard, M., Moncel, J., Parreau, A.: New results on variants of covering codes in Sierpiński graphs. Des. Codes Cryptogr. 69(2), 181–188 (2013) MathSciNetMATHView ArticleGoogle Scholar
  15. Gravier, S., Kovše, M., Parreau, A.: Generalized Sierpiński graphs, EuroComb’11 Budapest (2011). http://www.renyi.hu/conferences/ec11/posters/parreau.pdf
  16. Gutman, I., Polansky, O.: Mathematical Concepts in Organic Chemistry. Springer, Berlin (1986). http://dx.doi.org/10.1007/978-3-642-70982-1 MATHView ArticleGoogle Scholar
  17. Gutman, I., Trinajstic, N.: Total ϕ-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17, 535–538 (1972). https://doi.org/10.1016/0009-2614(72)85099-1 View ArticleGoogle Scholar
  18. Hansch, C., Leo, L.: Exploring QSAR fundamentals and applicability in chemistry and biology. J. Am. Chem. Soc. (1996) Google Scholar
  19. Hinz, A., Parisse, D.: The average eccentricity of Sierpiński graphs. Graphs Comb. 28(5), 671–686 (2012) MATHView ArticleGoogle Scholar
  20. Hinz, A.M., Auf Der Heide, C.H.: An efficient algorithm to determine all shortest paths in Sierpiński graphs. Discrete Appl. Math. 177, 111–120 (2014) MathSciNetMATHView ArticleGoogle Scholar
  21. Hinz, A.M., Parisse, D.: Coloring Hanoi and Sierpiński graphs. Discrete Math. 312, 1521–1535 (2012) MathSciNetMATHView ArticleGoogle Scholar
  22. Imran, M., Hafi, S., Gao, W., Farahani, M.R.: On topological properties of Sierpiński networks. Chaos Solitons Fractals 98, 199–204 (2017) MathSciNetMATHView ArticleGoogle Scholar
  23. Klavžar, S., Milutinović, U.: Graphs \(S(n, k)\) and a variant of the tower of hanoi problem. Czechoslov. Math. J. 47(1), 95–104 (1997) MathSciNetMATHGoogle Scholar
  24. Klavžar, S., Milutinović, U., Petr, C.: Bull. Aust. Math. Soc. 66(3), 369–384 (2002) View ArticleGoogle Scholar
  25. Klavžar, S., Mohar, B.: Crossing numbers of Sierpiński-like graphs. J. Graph Theory 50, 186–198 (2005) MathSciNetMATHView ArticleGoogle Scholar
  26. Malik, M.A., Imran, M.: On multiple Zagreb indices of T i O 2 $T_{i}O_{2}$ nanotubes. Acta Chim. Slov. (2015) Google Scholar
  27. Rostami, M., Shabanian, M., Moghanian, H.: Digest J. Nanomater. Biostruct. 7, 247–252 (2012) Google Scholar
  28. Todeschini, R., Consonni, V.: Handbook of Molecular Descriptors, Methods and Principles in Medicinal Chemistry. Wiley, New York (2008) Google Scholar
  29. Wiener, H.: Structural determination of paraffin boiling point. J. Am. Chem. Soc. 69, 17–20 (1947). https://doi.org/10.1021/ja01193a005 View ArticleGoogle Scholar
  30. Zhang, X., Yang, H., Gao, Y., Farahani, M.R.: Some degree-based topological indices of base-3 Sierpiński graphs. Science 5(3), 36–41 (2017) Google Scholar
  31. Zhou, B., Trinajstić, N.: On a novel connectivity index. J. Math. Chem. 46, 1252–1270 (2009) MathSciNetMATHView ArticleGoogle Scholar

Copyright

© The Author(s) 2019

Advertisement