Topological aspects of extended Sierpiński structures with help of underlying networks

Faiza Ishfaq; Muhammad Imran; Muhammad Faisal Nadeem

doi:10.1016/j.jksus.2022.102126

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Topological aspects of extended Sierpiński structures with help of underlying networks

Faiza Ishfaq^{^a}, Muhammad Imran^{^b,⁎}, Muhammad Faisal Nadeem^{^a}

a Department of Mathematics, COMSATS University Islamabad Lahore Campus, Lahore 54000, Pakistan

b Department of Mathematical Sciences, United Arab Emirates University, Al Ain, United Arab Emirates

⁎Corresponding author. m.imran658@uaeu.ac.ae (Muhammad Imran)

Received: 2022-1-6, Accepted: 2022-5-23,

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

Sierpiński networks are the most studied networks of fractal nature with applications in various fields of science. A generalized Sierpiński network is obtained by copying the base network, resulting in the self-similar network. The extended Sierpiński networks are obtained by introducing a new vertex in a generalized Sierpiński network and attaching this vertex with the extreme vertices. Certain network invariants are used to find thermodynamic properties, physio-chemical properties, and biological activities of chemical compounds. These network invariants play a dynamic role in QSAR/QSPR study. In this paper, we discussed Zagreb indices and forgotten topological index for extended Sierpiński networks by using any base network $H$ . Moreover, for the studied topological indices, we attained some bounds using different parameters i.e. order, size, maximum and minimum degrees of vertices in network $H$ .

Keywords

Zagreb indices

Forgotten index

Extended Sierpiński networks

Extremal networks

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1 Introduction

Sierpiński networks are the most studied networks similar to fractals. A fractal is a geometric structure that is self-similar throughout its structure. Fractal models are tremendously common since, nature is full of fractals, for example, plants, canals, coastlines, mountains, clouds, seashells, and tornadoes. Fractals help to study and comprehend key scientific ideas, such as how bacteria grow, freezing water patterns, and brain waves. Sierpiński and Sierpiński type networks are considered in fractal theory (Teplyaev, 1998). Klavžar and Milutinović showed that the Sierpiński networks are similar structure to the Tower of Hanoi (Klavžar and Milutinović, 1997). The Sierpiński networks have many attractive properties for instance coding and metric properties and play an important role in numerous areas of science i.e. dynamic systems, probability, psychology, biology, chemical graph theory, computer networking and physical sciences. For more detail see (Alquran et al., 2020; Naseem et al., 2021; Klavžar et al., 2002; Romik, 2006; Vecchia and Sanges, 1988 ).

The networks studied in this article assumed to be finite and simple. A network/graph $H = (V, E)$ is a collection of set of vertices $V (H)$ and set of edges $E (H)$ . The order of graph $H$ is the cardinality of its vertices, while cardinality of edges is called size and frequently denoted by p and q respectively. The degree of vertex v is known as the number of edges connected to that particular vertex and denoted by $d_{v}$ . A graph $H$ is known as complete if every two vertices are incident to each other. $δ (H)$ and $Δ (H)$ represent the minimum and maximum degree of a vertex in graph $H$ . If $δ (H) = Δ (H)$ = l, then $H$ is a l-regular graph. The path, star, cycle and complete graph of order p are represented by $P_{p}, S_{p}, C_{p}$ and $K_{p}$ .

In mathematical chemistry, chemical graph theory, and pharmaceutical industry, topological invariants are very important. The physio-chemical properties of chemical structures can be forecasted by using topological invariants. From the last few decades, several topological indices were established and examined in literature (Todeschini and Consonni, 2000), which are applied to attain the facts of numerous characteristics of organic materials which depend on their molecular structures. Wiener a chemist in 1947 introduced the first topological index in order to determine the boiling points of paraffins (Wiener, 1947).

Gutman et al. in Gutman and Trinajstić (1972) and Gutman et al. (1975) introduced the Zagreb indices, which are stated as $M_{1} (H) = \sum_{r \in V (H)} d_{H} {(r)}^{2} = \sum_{rs \in E (H)} (d_{H} (r) + d_{H} (s))$ $M_{2} (H) = \sum_{rs \in E (H)} (d_{H} (r) d_{H} (s))$

Furtula and Gutman (2015) proposed forgotten topological index, stated as $F (H) = \sum_{r \in V (H)} d_{H} {(r)}^{3} = \sum_{rs \in E (H)} (d_{H} {(r)}^{2} + d_{H} {(s)}^{2})$

For more detail on topological indices see Liu et al. (2019), Havare (2021), Akhter and Imran (2017), An and Das (2018), Che and Chen (2016), Cristea and Steinsky (2013), Gutman (2013), Horoldagva and Das (2015), Hua and Das (2013), Horoldagva et al. (2016), and Yoon and Kim (2006).

The generalized Sierpiński graph of dimension t is represented by $S (H, t)$ is a graph with vertex set $V^{t}$ , where $V = V (H)$ . The vertex set $V^{t}$ is the set of all words $v_{1} v_{2} \dots v_{t}$ of length t, where $v_{p} \in V, 1 ⩽ p ⩽ t$ , two vertices $u, w$ linked by an edge in $S (H, t)$ if and only if there is $i \in {1, 2, \dots, t}$ such that.

$u_{j} = w_{j}$ if $j < i$ .
$u_{i} \neq w_{i}$ and $u_{i}, w_{i} \in E (H)$ .
$u_{j} = w_{i}$ and $u_{i} = w_{j}$ if $j > i$ .

From above definition, it is clear that, if $uw \in E (S (H, t))$ then $rs \in E (H)$ and a word z such that $u = zrss \dots s$ and $w = zsrr \dots r$ . A vertex of the form $uu \dots u$ is known as extreme vertex and denoted by $\bar{w}$ . For a graph $H$ of order $p, S (H, t)$ has p extreme vertices. Moreover, extreme vertices have same degree in $S (H, t)$ as in base graph $H, d_{H} (u) + 1 = d_{S (H, t)} wuu \dots u$ and $d_{H} (w) + 1 = d_{S (H, t)} uww \dots w$ . Fig. 1 and Fig. 2 represents the generalized and extended Sierpiński graphs respectively, where extended Sierpiński graph is obtained by involving a new vertex x in generalized Sierpiński graph and joining it with extreme vertices. Extended Sierpiński graph is represented by $ES (H, t)$ .

Extended Sierpiński graph ES ( 2 , C 4 ) .

For $v \in V, d_{s} (H, t) (v) \in {d_{H} (v), d_{H} (v) + 1}$ , here $d_{H} (v)$ represents the degree of v in $H$ . For our convenience $d_{H} (v)$ is represented by $d_{v}$ in this article. Let $| d_{r}, d_{s} |_{ES (H, t)}$ is the number of copies of ${r, s}$ edge with degrees $d_{r}$ and $d_{s}$ in $ES (H, t)$ . For $r, s \in V (H), ◁ (r, s)$ represents the triangles of $H$ having r and s as its vertices, while $◁ (H)$ represents the number of triangles in $H$ . For $rs \in E (H)$ , we have $| N_{r} \cap N_{s} | = ◁ (r, s), | N_{r} \cup N_{s} | = d_{r} + d_{s} - ◁ (r, s)$ and $| N_{r} - N_{s} | = d_{r} - ◁ (r, s)$ . We used the function $ϕ_{p} (t) = 1 + p + p^{2} + \dots + p^{t - 1} = \frac{p^{t} - 1}{p - 1}$ for a graph of order p. Imran and Jamil (2020) calculate the constraints of generalized Sierpiński graphs. We will establish the results for topological properties of extended Sierpiński graph with any base graph $H$ . For these topological indices we will obtain some sharp bounds in terms of numerous parameters. In this article, we will select the first Zagreb, second Zagreb and forgotten indices to investigate the invariants of $ES (H, t)$ graphs. Following lemmas are helpful in finding the main results of the paper.

Lemma 1.1

Zhou (2004) Let $H$ be a graph without triangle having order p, size $q > 0$ . Then $M_{1} (H) ⩽ pq$ and equality holds if and only if $H$ is a complete bipartite graph.

Lemma 1.2

Zhou (2004) Let $H$ be a graph without triangle having size $q > 0$ . Then $M_{2} (H) ⩽ q^{2}$ and equality holds if and only if $H$ is a union of a complete bipartite graph and isolated vertices.

Lemma 1.3

Das (2003) Let p and $q > 0$ be vertices and edges respectively of a graph $H$ . Then $\frac{4 q^{2}}{p} ⩽ M_{1} (H) ⩽ (\frac{2 q}{p - 1} + p - 2)$ and left equality holds if and only if $H$ is a regular graph and right equality holds if and only if $H$ is $K_{p}, K_{1, p - 1}$ or $K_{1 \cup p - 1}$ .

Lemma 1.4

Zhou (2004) Let $q > 0$ be a size of a graph $H$ . Then $M_{2} (H) ⩽ q {(\sqrt{2} q + \frac{1}{4} - \frac{1}{2})}^{2}$ and equality holds if and only if $H$ is a union of a complete and isolated vertices.

Lemma 1.5

Estrada-Moreno and Rodríguez-Velázquez (2019) Let p be the order of a graph $H$ , for any edge rs and integer $t ⩾ 2$ , we have

$| d_{r}, d_{s} |_{S (H, t)} = p^{t - 2} (p - d_{r} - d_{s} + ◁ (r, s))$
$| d_{r} + 1, d_{s} |_{S (H, t)} = p^{t - 2} (d_{r} - ◁ (r, s)) - ϕ_{p} (t - 2) d_{s}$
$| d_{r}, d_{s} + 1 |_{S (H, t)} = p^{t - 2} (d_{s} - ◁ (r, s)) - ϕ_{p} (t - 2) d_{r}$
$| d_{r} + 1, d_{s} + 1 |_{S (H, t)} = p^{t - 2} (◁ (r, s) + 1) + ϕ_{p} (t - 2) (d_{r} + d_{s} + 1)$ .

2 Main results

In this part of paper, we obtained the Zagreb and forgotten topological indices for extended Sierpiński graph by considering any arbitrary graph $H$ . Furthermore, we also compute some bounds for $ES (H, t)$ . Here $d_{\bar{w}}$ represents the degree of extreme vertices and $d_{x}$ is the degree of new vertex which is introduced in generalized Sierpiński graph in order to obtain $ES (H, t)$ throughout this article. By using Lemma 1.5 we can deduce the following result for extended Sierpiński graph.

Lemma 2.1

Let p be the order of a graph $H$ , for any edge rs and integer $t ⩾ 2$ , we have

$| d_{r}, d_{s} |_{ES (H, t)} = p^{t - 2} (p - d_{r} - d_{s} + ◁ (r, s))$
$| d_{r} + 1, d_{s} |_{ES (H, t)} = p^{t - 2} (d_{r} - ◁ (r, s)) - ϕ_{p} (t - 2) d_{s} - 1$
$| d_{r}, d_{s} + 1 |_{ES (H, t)} = p^{t - 2} (d_{s} - ◁ (r, s)) - ϕ_{p} (t - 2) d_{r} - 1$
$| d_{r} + 1, d_{s} + 1 |_{ES (H, t)} = p^{t - 2} (◁ (r, s) + 1) + ϕ_{p} (t - 2) (d_{r} + d_{s} + 1) + 2$
$| d_{\bar{w}}, d_{x} |_{ES (H, t)} = p$ .

Theorem 2.2

Let p and q be vertices and edges of a graph $H$ . Then first Zagreb index of extended Sierpiński graph $ES (H, t)$ of the graph $H$ of dimension $t ⩾ 2$ is $M_{1} (ES (H, t)) = (ϕ_{p} (t) + ϕ_{p} (t - 1)) M_{1} (H) + 2 q (1 + ϕ_{p} (t - 1)) + p (d_{\bar{w}} + d_{x})$ .

Proof

Let p and q be vertices and edges of a graph $H$ . Then the first Zagreb index of $ES (H, t)$ can be defined as $\begin{matrix} M_{1} (ES (H, t)) = & \sum_{rs \in E (H)} \sum_{i, j = 0}^{1} | d_{r} + i, d_{s} + j | (d_{r} + i + d_{s} + j) \\ + | d_{\bar{w}}, d_{x} |_{ES (H, t)} (d_{\bar{w}} + d_{x}) \end{matrix}$

Now, by using Lemma 2.1 we have $\begin{matrix} M_{1} (ES (H, t)) = & \sum_{rs \in E (H)} [p^{t - 2} (p - d_{r} - d_{s} + ◁ (r, s)) (d_{r} + d_{s}) \\ + (p^{t - 2} (d_{r} - ◁ (r, s)) - ϕ_{p} (t - 2) d_{s} - 1) (d_{r} + d_{s} + 1) \\ + (p^{t - 2} (d_{s} - ◁ (r, s)) - ϕ_{p} (t - 2) d_{r} - 1) (d_{r} + d_{s} + 1) \\ + (p^{t - 2} (◁ (r, s) + 1) + ϕ_{p} (t - 2) (d_{r} + d_{s} + 1) + 2) (d_{r} + d_{s} + 2)] \\ + p (d_{\bar{w}} + d_{x}) \\ = & \sum_{rs \in E (H)} (p^{t - 1} + 2 p^{t - 2} + 2 ϕ_{p} (t - 2)) (d_{r} + d_{s}) \\ + \sum_{rs \in E (H)} 2 (1 + p^{t - 2} + ϕ_{p} (t - 2)) + p (d_{\bar{w}} + d_{x}) \\ = & (ϕ_{p} (t) + ϕ_{p} (t - 1)) M_{1} (H) + 2 q (1 + ϕ_{p} (t - 1)) + p (d_{\bar{w}} + d_{x}) . \end{matrix}$ □

From Lemma 1.3 we obtained the next result.

Corollary 2.3

Let p and $q > 0$ be vertices and edges respectively of a graph $H$ . Then $(ϕ_{p} (t) + ϕ_{p} (t - 1)) \frac{4 q^{2}}{p} + 2 q (1 + ϕ_{p} (t - 1)) + p (d_{\bar{w}} + d_{x}) ⩽ M_{1} (ES (H, t)) ⩽ \frac{ϕ_{p} (t)}{p - 1} (2 q + p^{2} - 3 p + 2) + \frac{ϕ_{p} (t - 1)}{p - 1} (2 pq + p^{2} - 3 p + 2) + 2 q (1 + ϕ_{p} (t - 1)) + p (d_{\bar{w}} + d_{x})$ .

The lower bound is obtained if $H$ is isomorphic to a regular graph and upper bound is obtained if $H$ is isomorphic to $K_{p}, K_{1, p - 1}$ or $K_{1 \cup p - 1}$ . Lemma 1.1 gives the result for the upper bound of $ES (H, t)$ .

Corollary 2.4

Let $H$ be a graph without triangle having order p, size $q > 0$ and $t ⩾ 1$ . Then $M_{1} (ES (H, t)) ⩽ (ϕ_{p} (t) + ϕ_{p} (t - 1)) pq + 2 q (1 + ϕ_{p} (t - 1)) + p (d_{\bar{w}} + d_{x})$ .

Corollary 2.5

Let $P_{p}, S_{p}, C_{p}$ and $K_{p}$ be path, star, cycle and complete graphs of order p. Then the first Zagreb index for extended Sierpiński graph with dimension $t ⩾ 1$ of these graphs is given as

$M_{1} (ES (P_{p}, t)) = ϕ_{p} (t) (4 p - 6) + ϕ_{p} (t - 1) (6 p - 8) + p^{2} + 5 p - 4$
$M_{1} (ES (S_{p}, t)) = ϕ_{p} (t) (p^{2} - p) + ϕ_{p} (t - 1) (p^{2} + p - 2) + p^{2} + 5 p - 4$
$M_{1} (ES (C_{p}, t)) = ϕ_{p} (t) (4 p) + ϕ_{p} (t - 1) (6 p) + p^{2} + 5 p$
$M_{1} (ES (K_{p}, t)) = ϕ_{p} (t) p {(p - 1)}^{2} + ϕ_{p} (t - 1) p^{2} (p - 1) + 3 p^{2} - p$ .

Proof

From Theorem 2.2, we have $M_{1} (ES (H, t)) = (ϕ_{p} (t) + ϕ_{p} (t - 1)) M_{1} (H) + 2 q (1 + ϕ_{p} (t - 1)) + p (d_{\bar{w}} + d_{x}) .$

Now, by replacing the value of $M_{1} (H), q$ and $p (d_{\bar{w}} + d_{x})$ by taking path, star, cycle and complete graph as a base graph in above equation, then we will obtain $\begin{matrix} M_{1} (ES (P_{p}, t)) = & (ϕ_{p} (t) + ϕ_{p} (t - 1)) (2 (3) + 4 (p - 3)) \\ + 2 (p - 1) (1 + ϕ_{p} (t - 1)) + 2 (p + 2) + (p - 2) (3 + p) \\ = & ϕ_{p} (t) (4 p - 6) + ϕ_{p} (t - 1) (6 p - 8) + p^{2} + 5 p - 4 \end{matrix}$ $\begin{matrix} M_{1} (ES (S_{p}, t)) = & (ϕ_{p} (t) + ϕ_{p} (t - 1)) (p - 1) (p) \\ + 2 (p - 1) (1 + ϕ_{p} (t - 1)) + (p + p) + (p - 1) (2 + p) \\ = & ϕ_{p} (t) (p^{2} - p) + ϕ_{p} (t - 1) (p^{2} + p - 2) + p^{2} + 5 p - 4 \end{matrix}$ $\begin{matrix} M_{1} (ES (C_{p}, t)) = & (ϕ_{p} (t) + ϕ_{p} (t - 1)) (4 P) + 2 (p) (1 + ϕ_{p} (t - 1)) + p (p + 3) \\ = & ϕ_{p} (t) (4 p) + ϕ_{p} (t - 1) (6 p) + p^{2} + 5 p . \end{matrix}$ $\begin{matrix} M_{1} (ES (K_{p}, t)) = & (ϕ_{p} (t) + ϕ_{p} (t - 1)) p {(p - 1)}^{2} \\ + p (p - 1) (1 + ϕ_{p} (t - 1)) + p (2 p) \\ = & ϕ_{p} (t) p {(p - 1)}^{2} + ϕ_{p} (t - 1) p^{2} (p - 1) + 3 p^{2} - p \end{matrix}$ □

Theorem 2.6

Let $H$ is a base graph with minimum and maximum degree $δ$ and $Δ$ respectively. Then for extended Sierpiński graphs, we have $2 q δ (ϕ_{p} (t) + ϕ_{p} (t - 1)) + 2 q (1 + ϕ_{p} (t - 1)) + p (d_{\bar{w}} + d_{x}) ⩽ M_{1} (ES (H, t)) ⩽ 2 q Δ (ϕ_{p} (t) + ϕ_{p} (t - 1)) + 2 q (1 + ϕ_{p} (t - 1)) + p (d_{\bar{w}} + d_{x})$ left equality holds if $H ≅ δ$ -regular graph and right equality holds if $H ≅ Δ$ -regular graph.

Proof

Let $H$ be a base graph having order p and size q. The first Zagreb index of $ES (H, t)$ can be stated as $\begin{matrix} M_{1} (ES (H, t)) = & \sum_{rs \in E (H)} \sum_{i, j = 0}^{1} | d_{r} + i, d_{s} + j | (d_{r} + i + d_{s} + j) \\ + | d_{\bar{w}}, d_{x} |_{ES (H, t)} (d_{\bar{w}} + d_{x}) \end{matrix}$

Now, by using Lemma 2.1 we have, $\begin{matrix} M_{1} (ES (H, t)) = & \sum_{rs \in E (H)} [p^{t - 2} (p - d_{r} - d_{s} + ◁ (r, s)) (d_{r} + d_{s}) \\ + (p^{t - 2} (d_{r} - ◁ (r, s)) - ϕ_{p} (t - 2) d_{s} - 1) (d_{r} + d_{s} + 1) \\ + (p^{t - 2} (d_{s} - ◁ (r, s)) - ϕ_{p} (t - 2) d_{r} - 1) (d_{r} + d_{s} + 1) \\ + (p^{t - 2} (◁ (r, s) + 1) + ϕ_{p} (t - 2) (d_{r} + d_{s} + 1) + 2) (d_{r} + d_{s} + 2)] \\ + p (d_{\bar{w}} + d_{x}) \end{matrix}$

Since, $δ (H) = δ$ is the minimum degree in graph $H$ , then we obtained $\begin{matrix} M_{1} (ES (H, t)) ⩾ & \sum_{rs \in E (H)} [p^{t - 2} (p - 2 δ + ◁ (r, s)) 2 δ \\ + (2 p^{t - 2} (δ - ◁ (r, s)) - 2 ϕ_{p} (t - 2) δ - 1) (2 δ + 1) \\ + (p^{t - 2} (◁ (r, s) + 1) + ϕ_{p} (t - 2) (2 δ + 1) + 2) (2 δ + 2)] \\ + p (d_{\bar{w}} + d_{x}) \\ = & 2 q δ (ϕ_{p} (t) + ϕ_{p} (t - 1)) + 2 q (1 + ϕ_{p} (t - 1)) + p (d_{\bar{w}} + d_{x}) \end{matrix}$ and equality holds if $H ≅ δ$ -regular graph

Since, $Δ (H) = Δ$ is the maximum degree in $H$ , then inequality becomes $\begin{matrix} M_{1} (ES (H, t)) ⩽ & \sum_{rs \in E (H)} [p^{t - 2} (p - 2 Δ + ◁ (r, s)) 2 Δ \\ + (2 p^{t - 2} (Δ - ◁ (r, s)) - 2 ϕ_{p} (t - 2) Δ - 1) (2 Δ + 1) \\ + (p^{t - 2} (◁ (r, s) + 1) + ϕ_{p} (t - 2) (2 Δ + 1) + 2) (2 Δ + 2)] \\ + p (d_{\bar{w}} + d_{x}) \\ ⩽ & 2 q Δ (ϕ_{p} (t) + ϕ_{p} (t - 1)) + 2 q (1 + ϕ_{p} (t - 1)) + p (d_{\bar{w}} + d_{x}) \end{matrix}$ and equality holds if $H ≅ Δ$ -regular graph. □

Corollary 2.7

Let $p ⩾ 3$ be the vertices of a regular graph $H$ . Then for extended Sierpiński graph, we have $ϕ_{p} (t) (4 p) + ϕ_{p} (t - 1) (6 p) + p^{2} + 5 p ⩽ M_{1} (ES (H, t)) ⩽ ϕ_{p} (t) p {(p - 1)}^{2} + ϕ_{p} (t - 1) p^{2} (p - 1) + 3 p^{2} - p$ the left equality holds if $H ≅ C_{p}$ and right equality holds if $H ≅ K_{p}$ .

Now, in next theorem we compute the formula of second zagreb index for extended Sierpiński graph.

Theorem 2.8

Let $ES (H, t)$ be extended Sierpiński graph with dimension $t ⩾ 2$ , where $H$ having p vertices and q edges. Then second Zagreb index of $ES (H, t)$ is $M_{2} (ES (H, t)) = (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) M_{2} (H) + (ϕ_{p} (t - 1) + ϕ_{p} (t - 2) + 1) M_{1} (H) + q (2 + ϕ_{p} (t - 1)) + p (d_{\bar{w}} \times d_{x}) + p^{t - 2} \sum_{rs \in E (H)} ◁ (r, s)$ .

Proof

Let $H$ be a graph having p vertices and q edges. The second Zagreb index of $ES (H, t)$ can be stated as $\begin{matrix} M_{2} (ES (H, t)) = & \sum_{rs \in E (H)} \sum_{i, j = 0}^{1} | d_{r} + i, d_{s} + j | (d_{r} + i) (d_{s} + j) \\ + | d_{\bar{w}}, d_{x} |_{ES (H, t)} (d_{\bar{w}} \times d_{x}) \end{matrix}$

Now, by using Lemma 2.1 we have $\begin{matrix} M_{2} (ES (H, t)) = & \sum_{rs \in E (H)} [p^{t - 2} (p - d_{r} - d_{s} + ◁ (r, s)) (d_{r} \times d_{s}) \\ + (p^{t - 2} (d_{r} - ◁ (r, s)) - ϕ_{p} (t - 2) d_{s} - 1) (d_{r} + 1) (d_{s}) \\ + (p^{t - 2} (d_{s} - ◁ (r, s)) - ϕ_{p} (t - 2) d_{r} - 1) (d_{r}) (d_{s} + 1) \\ + (p^{t - 2} (◁ (r, s) + 1) + ϕ_{p} (t - 2) (d_{r} + d_{s} + 1) + 2) (d_{r} + 1) (d_{s} + 1)] \\ + p (d_{\bar{w}} \times d_{x}) \\ = & \sum_{rs \in E (H)} [(p^{t - 1} + p^{t - 2} + ϕ_{p} (t - 2) + 2 ϕ_{p} (t - 2) + 2 p^{t - 2}) (d_{r} \times d_{s})] \\ + \sum_{rs \in E (H)} [(2 + p^{t - 2} + ϕ_{p} (t - 2)) + (p^{t - 2} + 2 ϕ_{p} (t - 2) + 1) (d_{r} + d_{s})] \\ + p^{t - 2} \sum_{rs \in E (H)} ◁ (r, s) + p (d_{\bar{w}} \times d_{x}) \\ = & (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) M_{2} (H) + (ϕ_{p} (t - 1) + ϕ_{p} (t - 2) + 1) M_{1} (H) \\ + q (2 + ϕ_{p} (t - 1)) + p (d_{\bar{w}} \times d_{x}) + p^{t - 2} \sum_{rs \in E (H)} ◁ (r, s) . \end{matrix}$ □

Corollary 2.9

Let $P_{p}, S_{p}, C_{p}$ and $K_{p}$ be path, star, cycle and complete graphs of order p. Then second Zagreb index for extended Sierpiński graph with dimension $t ⩾ 1$ of these graphs is given as

$M_{2} (ES (P_{p}, t)) = ϕ_{p} (t) (4 p - 8) + ϕ_{p} (t - 1) (13 p - 23) + ϕ_{p} (t - 2) (4 p - 6) + 3 p^{2} + 4 p - 8$

$M_{2} (ES (S_{p}, t)) = ϕ_{p} (t) {(p - 1)}^{2} + ϕ_{p} (t - 1) (3 p^{2} - 4 p + 1) + ϕ_{p} (t - 2) (p^{2} - p) + 4 p^{2} - p - 2; p ⩾ 4$

$M_{2} (ES (C_{p}, t)) = (ϕ_{p} (t) + ϕ_{p} (t - 2)) (4 p) + ϕ_{p} (t - 1) (13 p) + 3 p^{2} + 6 p$

$M_{2} (ES (K_{p}, t)) = ϕ_{p} (t) \frac{p {(p - 1)}^{3}}{2} + \frac{1}{2} ϕ_{p} (t - 1) (2 p^{4} - 4 p^{3} + 3 p^{2} - p) + ϕ_{p} (t - 2) p {(p - 1)}^{2} + 2 p^{3} - p^{2} + p^{t - 2} (\frac{p^{3} - 3 p^{2} + 2 p}{2})$ .

Proof

From Theorem 2.8, we have $\begin{matrix} M_{2} (ES (H, t)) = & (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) M_{2} (H) + (ϕ_{p} (t - 1) + ϕ_{p} (t - 2) + 1) M_{1} (H) \\ + q (2 + ϕ_{p} (t - 1)) + p (d_{\bar{w}} \times d_{x}) + p^{t - 2} \sum_{rs \in E (H)} ◁ (r, s) \\ . \end{matrix}$

Now, by replacing the value of $M_{1} (H), M_{2} (H), q$ and $p (d_{\bar{w}} \times d_{x})$ and $◁ (r, s)$ by taking path, star, cycle and complete graph as a base graph in above equation, then we will obtain $\begin{matrix} M_{2} (ES (P_{p}, t)) = & (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) (4 + (p - 3) 4) + (ϕ_{p} (t - 1) + ϕ_{p} (t - 2) + 1) (6 + 4 (p - 3)) \\ + & (p - 1) (2 + ϕ_{p} (t - 1)) + 4 p + (p - 2) 3 p \\ = & ϕ_{p} (t) (4 p - 8) + ϕ_{p} (t - 1) (13 p - 23) + ϕ_{p} (t - 2) (4 p - 6) + 3 p^{2} + 4 p - 8 \end{matrix}$ $\begin{matrix} M_{2} (ES (S_{p}, t)) = & (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) ({(p - 1)}^{2}) + (ϕ_{p} (t - 1) + ϕ_{p} (t - 2) + 1) (p (p - 1)) \\ + (p - 1) (2 + ϕ_{p} (t - 1)) + p^{2} + (p - 1) 2 p \\ = & ϕ_{p} (t) {(p - 1)}^{2} + ϕ_{p} (t - 1) (3 p^{2} - 4 p + 1) + ϕ_{p} (t - 2) (p^{2} - p) + 4 p^{2} - p - 2 \end{matrix}$ $\begin{matrix} M_{2} (ES (C_{p}, t)) & = (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) (4 p) + (ϕ_{p} (t - 1) + ϕ_{p} (t - 2) + 1) (4 p) \\ + (p) (2 + ϕ_{p} (t - 1)) + p (p - 1) \\ = (ϕ_{p} (t) + ϕ_{p} (t - 2)) (4 p) + ϕ_{p} (t - 1) (13 p) + 3 p^{2} + 6 p \end{matrix}$ $\begin{matrix} M_{2} (ES (K_{p}, t)) & = (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) \frac{p {(p - 1)}^{3}}{2} + (ϕ_{p} (t - 1) + ϕ_{p} (t - 2) + 1) (p {(p - 1)}^{2}) \\ + (2 + ϕ_{p} (t - 1)) \frac{p (p - 1)}{2} + p^{3} + 3 p^{t - 2} ◁ (H) \\ = (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) \frac{p {(p - 1)}^{3}}{2} + (ϕ_{p} (t - 1) + ϕ_{p} (t - 2) + 1) (p {(p - 1)}^{2}) \\ + (2 + ϕ_{p} (t - 1)) \frac{p (p - 1)}{2} + p^{3} + p^{t - 2} (\frac{p^{3} - 3 p^{2} + 2 p}{2}) \\ = ϕ_{p} (t) \frac{p {(p - 1)}^{3}}{2} + \frac{1}{2} ϕ_{p} (t - 1) (2 p^{4} - 4 p^{3} + 3 p^{2} - p) + ϕ_{p} (t - 2) p {(p - 1)}^{2} \\ + 2 p^{3} - p^{2} + p^{t - 2} (\frac{p^{3} - 3 p^{2} + 2 p}{2}) \end{matrix}$ □

Theorem 2.10

Let $H$ is a base graph with minimum and maximum degree $δ$ and $Δ$ respectively. Then for extended Sierpiński graph, we have $μ (p, δ) + p^{t - 2} \sum_{rs \in E (H)} ◁ (r, s) + p (d_{\bar{w}} \times d_{x}) ⩽ M_{2} (ES (H, t)) ⩽ μ (p, Δ) + p^{t - 2} \sum_{rs \in E (H)} ◁ (r, s) + p (d_{\bar{w}} \times d_{x})$ where $μ (p, δ) = q δ^{2} (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) + 2 q δ (ϕ_{p} (t - 1) + ϕ_{p} (t - 2) + 1) + q (ϕ_{p} (t - 1) + 2)$ left equality holds if $H ≅ δ$ -regular graph and right and only if $H ≅ Δ$ -regular graph.

Proof

Let p and q are the order and size respectively of a graph $H$ . Then second Zagreb index of $ES (H, t)$ can be stated as $\begin{matrix} M_{2} (ES (H, t)) = & \sum_{rs \in E (H)} \sum_{i, j = 0}^{1} | d_{r} + i, d_{s} + j | (d_{r} + i) (d_{s} + j) \\ + | d_{\bar{w}}, d_{x} |_{ES (H, t)} (d_{\bar{w}} \times d_{x}) \end{matrix}$

Since, $δ (H) = δ$ is the minimum degree of $H$ . Then we obtained $\begin{matrix} M_{2} (ES (H, t)) ⩾ & \sum_{rs \in E (H)} [p^{t - 2} (p - 2 δ + ◁ (r, s)) δ^{2} \\ + 2 (p^{t - 2} (δ - ◁ (r, s)) - 2 ϕ_{p} (t - 2) δ - 1) (δ^{2} + δ) \\ + (p^{t - 2} (◁ (r, s) + 1) + ϕ_{p} (t - 2) (2 δ + 1) + 2) (δ^{2} + 2 δ + 1)] \\ + p (d_{\bar{w}} \times d_{x}) \\ = & q δ^{2} (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) + 2 q δ (ϕ_{p} (t - 1) + ϕ_{p} (t - 2) + 1) + q (ϕ_{p} (t - 1) + 2) \\ + p (d_{\bar{w}} \times d_{x}) + p^{t - 2} \sum_{rs \in E (H)} ◁ (r, s) \end{matrix}$ and equality holds if $H ≅ δ$ -regular graph.

As $Δ (H) = Δ$ is the maximum degree of $H$ . Then we obtained $\begin{matrix} M_{2} (ES (H, t)) ⩽ & \sum_{rs \in E (H)} [p^{t - 2} (p - 2 Δ + ◁ (r, s)) Δ^{2} \\ + (2 p^{t - 2} (Δ - ◁ (r, s)) - 2 ϕ_{p} (t - 2) Δ - 1) (Δ^{2} + Δ) \\ + (p^{t - 2} (◁ (r, s) + 1) + ϕ_{p} (t - 2) (2 Δ + 1) + 2) (Δ^{2} + 2 Δ + 1)] \\ + p (d_{\bar{w}} \times d_{x}) \\ = & q Δ^{2} (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) + 2 q Δ (ϕ_{p} (t - 1) + ϕ_{p} (t - 2) + 1) \\ + q (ϕ_{p} (t - 1) + 2) + p (d_{\bar{w}} \times d_{x}) + p^{t - 2} \sum_{rs \in E (H)} ◁ (r, s) \end{matrix}$ and equality holds if $H ≅ Δ$ -regular graph. □

If $H$ is without triangle, then above result becomes as follow.

Corollary 2.11

Let $H$ be a graph without triangle. Then $μ (p, δ) + p (d_{\bar{w}} \times d_{x}) ⩽ M_{2} (ES (H, t)) ⩽ μ (p, Δ) + p (d_{\bar{w}} \times d_{x})$ .

Corollary 2.12

Let $p ⩾ 4$ be the order of a connected regular graph $H$ . Then $(ϕ_{p} (t) + ϕ_{p} (t - 2)) (4 p) + ϕ_{p} (t - 1) (13 p) + 3 p^{2} + 6 p ⩽ M_{2} (ES (G, t)) ⩽ ϕ_{p} (t) \frac{p {(p - 1)}^{3}}{2} + \frac{1}{2} ϕ_{p} (t - 1) (2 p^{4} - 4 p^{3} + 3 p^{2} - p) + ϕ_{p} (t - 2) p {(p - 1)}^{2} + 2 p^{3} - p^{2} + p^{t - 2} (\frac{p^{3} - 3 p^{2} + 2 p}{2})$ . The left equality holds if $H ≅ C_{p}$ and right equality holds if $H ≅ K_{p}$ .

The following theorem gives the exact formula of forgotten index of $ES (H, t)$ .

Theorem 2.13

Let $ES (H, t)$ be extended Sierpiński graph of dimension $t ⩾ 2$ of base graph $H$ with p vertices and q edges. Then the forgotten topological index of $ES (H, t)$ is $F (ES (H, t)) = (3 ϕ_{p} (t - 1) + 2) M_{1} (H) + (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) F (H) + 2 q (ϕ_{p} (t - 1) + 1) + p (d_{\bar{w}}^{2} + d_{x}^{2})$ .

Proof

Let p and q be order and size of a graph $H$ . Then forgotten topological index of $ES (H, t)$ can be defined as $\begin{matrix} F (ES (H, t)) = & \sum_{rs \in E (H)} \sum_{i, j = 0}^{1} | d_{r} + i, d_{s} + j | {(d_{r} + i)}^{2} + {(d_{s} + j)}^{2} \\ + | d_{\bar{w}}, d_{x} |_{ES (H, t)} (d_{\bar{w}}^{2} + d_{x}^{2}) \end{matrix}$

Now, by using Lemma 2.1 we have $\begin{matrix} F (ES (H, t)) = & \sum_{rs \in E (H)} [p^{t - 2} (p - d_{r} - d_{s} + ◁ (r, s)) (d_{r}^{2} + d_{s}^{2}) \\ + (p^{t - 2} (d_{r} - ◁ (r, s)) - ϕ_{p} (t - 2) d_{s} - 1) ({(d_{r} + 1)}^{2} + d_{s}^{2}) \\ + (p^{t - 2} (d_{s} - ◁ (r, s)) - ϕ_{p} (t - 2) d_{r} - 1) (d_{r}^{2} + {(d_{s} + 1)}^{2}) \\ + (p^{t - 2} (◁ (r, s) + 1) + ϕ_{p} (t - 2) (d_{r} + d_{s} + 1) + 2) ({(d_{r} + 1)}^{2} + {(d_{s} + 1)}^{2})] \\ + p (d_{\bar{w}}^{2} + d_{x}^{2}) \\ = & \sum_{rs \in E (H)} [(ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) (d_{r}^{2} + d_{s}^{2}) + (3 ϕ_{p} (t - 1) + 2) (d_{r} + d_{s}) \\ + 2 (ϕ_{p} (t - 1) + 1)] + p (d_{\bar{w}}^{2} + d_{x}^{2}) \\ = & (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) F (H) + (3 ϕ_{p} (t - 1) + 2) M_{1} (H) + 2 q (ϕ_{p} (t - 1) + 1) \\ + p (d_{\bar{w}}^{2} + d_{x}^{2}) . \end{matrix}$ □

Corollary 2.14

Let $P_{p}, S_{p}, C_{p}$ and $K_{p}$ be path, star, cycle and complete graphs of order p. Then forgotten topological index for extended Sierpiński graph with dimension $t ⩾ 1$ of these graphs is given as

$F (ES (P_{p}, t)) = ϕ_{p} (t) (8 p - 14) + ϕ_{p} (t - 1) (13 p - 48) + p^{3} + 19 p - 24$

$F (ES (S_{p}, t)) = ϕ_{p} (t) (p^{3} - 3 p^{2} + 4 p - 2) + ϕ_{p} (t - 1) (2 p^{3} - 3 p^{2} + 7 p - 6) + p^{3} + 3 p^{2} + 4 p - 6$

$F (ES (C_{p}, t)) = ϕ_{p} (t) 8 p + ϕ_{p} (t - 1) (30 p) + p^{3} + 19 p$

$F (ES (K_{p}, t)) = ϕ_{p} (t) p {(p - 1)}^{3} + ϕ_{p} (t - 1) (2 p^{4} - 3 p^{3} + p^{2}) + 4 p^{3} - 3 p^{2} + p$ .

Proof

From Theorem 2.13, we have $F (ES (H, t)) = (3 ϕ_{p} (t - 1) + 2) M_{1} (H) + (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) F (H) + 2 q (ϕ_{p} (t - 1) + 1) + p (d_{\bar{w}}^{2} + d_{x}^{2}) .$

Now, by replacing the value of $M_{1} (H), q, F (H)$ and $p (d_{\bar{w}}^{2} + d_{x}^{2})$ by taking path, Star, Cycle and complete graph as a base graph in above equation, then we will obtain $\begin{matrix} F (ES (P_{p}, t)) & = (3 ϕ_{p} (t - 1) + 2) (4 p - 6) + (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) (8 p - 14) \\ + 2 (p - 1) (ϕ_{p} (t - 1) + 1) + 2 {(2 + p)}^{2} + {(p - 2)}^{2} {(p + 3)}^{2} \\ = ϕ_{p} (t) (8 p - 14) + ϕ_{p} (t - 1) (13 p - 48) + p^{3} + 19 p - 24 \\ F (ES (S_{p}, t)) & = (3 ϕ_{p} (t - 1) + 2) p (p - 1) + (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) (p^{3} - 3 p^{2} + 4 p - 2) \\ + 2 (p - 1) (ϕ_{p} (t - 1) + 1) + 2 p^{2} + (p - 1) (2^{2} + p^{2}) \\ = ϕ_{p} (t) (p^{3} - 3 p^{2} + 4 p - 2) + ϕ_{p} (t - 1) (2 p^{3} - 3 p^{2} + 7 p - 6) + p^{3} + 3 p^{2} + 4 p - 6 \\ F (ES (C_{p}, t)) & = (3 ϕ_{p} (t - 1) + 2) (4 p) + (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) (8 p) \\ + 2 p (ϕ_{p} (t - 1) + 1) + p (9 + p^{2}) \\ = ϕ_{p} (t) 8 p + ϕ_{p} (t - 1) (30 p) + p^{3} + 19 p \\ F (ES (K_{p}, t)) & = (3 ϕ_{p} (t - 1) + 2) p {(p - 1)}^{2} + (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) p {(p - 1)}^{3} \\ + p (p - 1) (ϕ_{p} (t - 1) + 1) + p (2 p^{2}) \\ = ϕ_{p} (t) p {(p - 1)}^{3} + ϕ_{p} (t - 1) (2 p^{4} - 3 p^{3} + p^{2}) + 4 p^{3} - 3 p^{2} + p \end{matrix}$ □

Theorem 2.15

If $H$ is a base graph, where $δ$ and $Δ$ are minimum and maximum degrees respectively. Then for extended Sierpiński graph, we have

$2 q δ^{2} (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) + 2 q δ (3 ϕ_{p} (t - 1) + 2) + 2 q (ϕ_{p} (t - 1) + 1) + p (d_{\bar{w}}^{2} + d_{x}^{2}) ⩽ F (ES (H, t)) ⩽ 2 q Δ^{2} (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) + 2 q Δ (3 ϕ_{p} (t - 1) + 2) + 2 q (ϕ_{p} (t - 1) + 1) + p (d_{\bar{w}}^{2} + d_{x}^{2})$ left equality holds if $H ≅ δ$ -regular graph and right equality holds if $H ≅ Δ$ -regular graph.

Proof

Let p and q be order and size respectively of a graph $H$ . Then forgotten topological index of $ES (H, t)$ can be stated as $\begin{matrix} F (ES (H, t)) = & \sum_{rs \in E (G)} \sum_{i, j = 0}^{1} | d_{r} + i, d_{s} + j | ({(d_{r} + i)}^{2} + {(d_{s} + j)}^{2}) \\ + | d_{\bar{w}}, d_{x} |_{ES (H, t)} (d_{\bar{w}}^{2} + d_{x}^{2}) \end{matrix}$

Now, by using Lemma 2.1 we have $\begin{matrix} F (ES (H, t)) = & \sum_{rs \in E (G)} [p^{t - 2} (p - d_{r} - d_{s} + ◁ (r, s)) (d_{r}^{2} + d_{s}^{2}) \\ + (p^{t - 2} (d_{r} - ◁ (r, s)) - ϕ_{p} (t - 2) d_{s} - 1) ({(d_{r} + 1)}^{2} + d_{s}^{2}) \\ + (p^{t - 2} (d_{s} - ◁ (r, s)) - ϕ_{p} (t - 2) d_{r} - 1) (d_{r}^{2} + {(d_{s} + 1)}^{2}) \\ + (p^{t - 2} (◁ (r, s) + 1) + ϕ_{p} (t - 2) (d_{r} + d_{s} + 1) + 2) ({(d_{r} + 1)}^{2} + {(d_{s} + 1)}^{2})] \\ + p (d_{\bar{w}}^{2} + d_{x}^{2}) \end{matrix}$

As $δ (H) = δ$ is the minimum degree of graph $H$ . Then we have $\begin{matrix} F (ES (H, t)) ⩾ & \sum_{rs \in E (H)} [p^{t - 2} (p - 2 δ + ◁ (r, s)) 2 δ^{2} \\ + 2 (p^{t - 2} (δ - ◁ (r, s)) - 2 ϕ_{p} (t - 2) δ - 1) (2 δ^{2} + 2 δ + 1) \\ + (p^{t - 2} (◁ (r, s) + 1) + ϕ_{p} (t - 2) (2 δ + 1) + 2) (2 δ^{2} + 4 δ + 2)] \\ + p (d_{\bar{w}}^{2} + d_{x}^{2}) \\ = & 2 q δ^{2} (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) + 2 q δ (3 ϕ_{p} (t - 1) + 2) + 2 q (ϕ_{p} (t - 1) + 1) \\ + p (d_{\bar{w}}^{2} + d_{x}^{2}) \end{matrix}$ and equality holds if $H ≅ δ$ -regular graph

As $Δ (H) = Δ$ is the maximum degree of graph $H$ . Then we obtained $\begin{matrix} F (ES (H, t)) ⩽ & \sum_{rs \in E (H)} [p^{t - 2} (p - 2 Δ + ◁ (r, s)) 2 Δ^{2} \\ + 2 (p^{t - 2} (Δ - ◁ (r, s)) - 2 ϕ_{p} (t - 2) Δ - 1) (2 Δ^{2} + 2 Δ + 1) \\ + (p^{t - 2} (◁ (r, s) + 1) + ϕ_{p} (t - 2) (2 Δ + 1) + 2) (2 Δ^{2} + 4 Δ + 2)] \\ + p (d_{\bar{w}}^{2} + d_{x}^{2}) \\ = & 2 q Δ^{2} (ϕ_{p} (t) + 2 ϕ_{p} (t - 1)) + 2 q Δ (3 ϕ_{p} (t - 1) + 2) + 2 q (ϕ_{p} (t - 1) + 1) \\ + p (d_{\bar{w}}^{2} + d_{x}^{2}) \end{matrix}$ and equality holds if $H ≅ Δ$ -regular graph. □

Corollary 2.16

Let $p ⩾ 3$ be the order of a base graph $H$ . Then

$ϕ_{p} (t) 8 p + ϕ_{p} (t - 1) 30 p + p^{3} + 19 p ⩽ F (ES (H, t)) ⩽ ϕ_{p} (t) p {(p - 1)}^{3} + ϕ_{p} (t - 1) (2 p^{4} - 3 p^{3} + p^{2}) + 4 p^{3} - 3 p^{2} + p$ left equality holds if $H ≅ C_{p}$ and right equality holds if $H ≅ K_{p}$ .

3 Conclusion

The extended Sierpiński graphs are obtained by introducing a new vertex in generalized Sierpiński graph and attached this vertex with extreme vertices. In this paper, we have compute the Zagreb and forgotten invariants for extended Sierpiński graphs using any base graph $H$ . Moreover, for these topological indices of extended Sierpiński graph, we attained some sharp bounds by applying numerous parameters. In future, we want to extend this work by applying other topological indices on extended Sierpiński graphs and attained the fruitful results.

Data availability statements

All the data used to finding the results is included in the manuscript.

Acknowledgments

This research is supported by the University program of Advanced Research (UPAR) and UAEU-AUA grants of United Arab Emirates University (UAEU) via Grant No.G00003271 and Grant No. G00003461.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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[1] Akhter S., Imran M., . Computing the forgotten topological index of four operations on graphs. AKCE Int. J. Graphs Combinat.. 2017;14(1):70-79.
[Google Scholar]

[2] Alquran M., Jaradat I., Abdel-Muhsen R., . Embedding (3+1)-dimensional diffusion, telegraph, and Burger’s equations into fractal 2D and 3D spaces: An analytical study. J. King Saud Univ.- Sci.. 2020;32(1):349-355.
[Google Scholar]

[3] An M., Das K.C., . First Zagreb index, $k$ -connectivity, beta-deficiency and $k$ -hamiltonicity of graphs. MATCH Commun. Math. Comput. Chem.. 2018;80:141-151.
[Google Scholar]

[4] Che Z., Chen Z., . Lower and upper bounds of the forgotten topological index. MATCH Commun. Math. Comput. Chem.. 2016;76:635-648.
[Google Scholar]

[5] Cristea L.L., Steinsky B., . Distances in Sierpiński graphs and on the Sierpiński gasket. Aequationes Mathematicae. 2013;85(3):201-219.
[Google Scholar]

[6] Das K.C., . Sharp bounds for the sum of the squares of the degrees of a graph. Kragujevac J. Math.. 2003;25:31-49.
[Google Scholar]

[7] Estrada-Moreno A., Rodríguez-Velázquez J.A., . On the General Randić index of polymeric networks modelled by generalized Sierpiński graphs. Discrete Appl. Math.. 2019;263:140-151.
[Google Scholar]

[8] Furtula B., Gutman I., . A forgotten topological index. J. Math. Chem.. 2015;53(4):1184-1190.
[Google Scholar]

[9] Gutman I., . Degree-based topological indices. Croat. Chem. Acta. 2013;86(4):351-361.
[Google Scholar]

[10] Gutman I., Trinajstić N., . Graph theory and molecular orbitals, Total $φ$ -electron energy of alternant hydrocarbons. Chem. Phys. Lett.. 1972;17(4):535-538.
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Topological aspects of extended Sierpiński structures with help of underlying networks

Abstract

Keywords

Zagreb indices

Forgotten index

Extended Sierpiński networks

Extremal networks

1 Introduction

2 Main results

3 Conclusion

Data availability statements

Acknowledgments

Declaration of Competing Interest

References

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