Entropies and Degree-Based Topological Indices of Generalized Sierpiński Graphs

Abstract

Fractals are geometric patterns that appear self-similar across all length scales and are constructed by repeating a single unit on a regular basis. Entropy, as a core thermodynamic function, is an extension based on information theory (such as Shannon entropy) that is used to describe the topological structural complexity or degree of disorder in networks. Topological indices, as graph invariants, provide quantitative descriptors for characterizing global structural properties. In this paper, we investigate two types of generalized Sierpiński graphs constructed on the basis of different seed graphs, and employ six topological indices—the first Zagreb index, the second Zagreb index, the forgotten index, the augmented Zagreb index, the Sombor index, and the elliptic Sombor index—to analyze the corresponding entropy. We utilize the method of edge partition based on vertex degrees and derive analytical formulations for the first Zagreb entropy, the second Zagreb entropy, the forgotten entropy, the augmented Zagreb entropy, the Sombor entropy, and the elliptic Sombor entropy. This research approach, which integrates entropy with Sierpiński network characteristics, furnishes novel perspectives and instrumental tools for addressing challenges in chemical graph theory, computer networks, and other related fields.

1. Introduction

Information entropy, a fundamental concept in information theory introduced by Shannon [1], quantifies uncertainty in a system. It is widely applied across various scientific domains, including chemistry, complex systems analysis, and network science. In graph theory, entropy is a key metric for analyzing the complexity and connectivity of network structures [2]. An extensive overview of graph entropy measures can be found in [3]. The entropy of an edge-weighted graph was introduced in [4]. Topological indices, as graph invariants, provide quantitative descriptors for characterizing global structural properties. For molecular compounds, topological indices offer insights into their chemical and physical properties, as well as their biological activity [5,6,7]. The degree-based entropy, proposed by Manzoor et al. [8], extends Shannon entropy to topological indices, enhancing the characterization of graph-based structures. In recent years, entropy related to topological indices of various networks has been investigated, such as coronene fractal structures [9], non-Kekulean benzenoid graphs [10], boron networks and polyphenylene networks [11].

Sierpiński gasket is a classic fractal structure [12]. Graphs of “Sierpiński type” are of significant importance across various fields of mathematics and also in several other scientific areas. In 1997, Klavžar and Milutinović [13] introduced the graphs $S(k,t)$, which generalize the graphs associated with the Tower of Hanoi problem. These graphs were later named Sierpiński graphs in [14]. Since their inception, Sierpiński graphs within the mathematical domain have been extensively studied, for instance, codes [14,15], Lipscomb space [16,17], chromatic number and chromatic index [18], crossing number [19], metric properties [20], and average eccentricity [21]. Furthermore, Sierpiński graphs have been extended to arbitrary graphs, giving rise to what are now known as generalized Sierpiński graphs [22]. For this graph class, Chanda and Iyer [23] investigated the Sombor index, Javaid et al. [24] derived bounds for four indices, and Alizadeh et al. [25] examined the corresponding metric properties. Rodríguez-Velázquez et al. [26] studied the Randić index, which was later extended to the generalized Randić index by [27].

All graphs and networks considered in this study are simple, undirected graphs. Let $V\left(G\right)$ and $E\left(G\right)$ represent the vertex set and edge set of graph G, respectively. The degree of a vertex i is the number of edges incident to i, denoted by $d_i$. Let $K_n$ denote the complete graph with n vertices, where any two vertices have an edge between them. The standard notations are mainly followed in [28]. We now introduce the concept of generalized Sierpiński graphs. Given an integer t and a graph G with vertex set $V\left(G\right)=\{1,2,\dots ,n\}$, the graph $S(G,t)$, referred to as the generalized Sierpiński graph of G [22], is constructed as follows:

•

$V\left(S(G,t)\right)=\{x_1{x}_2\cdots x_t\mid x_i\in V\left(G\right)$ for $1\le i\le t\}$.

•

$E\left(S\right(G,t\left)\right)$: For two vertices $x=x_1{x}_2\cdots x_t$ and $y=y_1{y}_2\cdots y_t$ in $V\left(S\right(G,t\left)\right)$, $\{x,y\}\in E\left(S\right(G,t\left)\right)$ if there exists $p\in \{1,2,\dots ,t\}$ such that

(i)

For all $q<p$, $x_q=y_q$;

(ii)

$x_p\ne y_p$ and $\{x_p,y_p\}\in E\left(G\right)$;

(iii)

For all $q>p$, $x_q=y_p$ and $y_q=x_p$.

Alternatively, an edge $\{x,y\}$ in $S(G,t)$ can be described by an edge $\{x_i,y_i\}$ in G and a prefix word $w=x_1{x}_2\cdots x_{i-1}$ such that x can be formed by concatenating w with the alternating sequence $x_i{y}_i\cdots y_i$ , and similarly for y with w followed by $y_i{x}_i\cdots x_i$. The graph G is termed the seed graph for $S(G,t)$.

,In general, $S(G,t)$ is generated recursively from G as follows: $S(G,1)=G$ and for $t\ge 2$, we create n copies of $S(G,t-1)$, prefixing the letter h to the label of each vertex in the copy associated with h. Then, for each $\{g,h\}$ of G, connect $ghh\cdots h$ and $hgg\cdots g$. For example, when M as shown in Figure 1, the generalized Sierpiński graphs $S(M,1),S(M,2)$, and $S(M,3)$, along with their corresponding vertex labelings, are shown in Figure 2. When $N$ as shown in Figure 1, the generalized Sierpiński graphs $S$, $S$, $S$ are displayed in Figure 3.

Figure 1. Graphs M and N: (a) Graph M, formed by removing one edge from the complete graph $K_4$. (b) Graph N, a graph with 5 vertices and 6 edges.

Figure 2. The generalized Sierpiński graphs $S(M,1),S(M,2)$ and $S(M,3)$, where M is depicted in Figure 1.

Figure 3. The generalized Sierpiński graphs $S(N,1),S(N,2)$ and $S(N,3)$, where N is a house with 5 vertices and 6 edges.

Next, we introduce the definitions of six degree-based topological indices. The first and second Zagreb indices [29] were defined as $M_1\left(G\right)={\sum }_{uv\in E\left(G\right)}(d_u+d_v)$ and $M_2\left(G\right)={\sum }_{uv\in E\left(G\right)}{d}_u{d}_v$. The forgotten index [30] was given by $F\left(G\right)={\sum }_{uv\in E\left(G\right)}(d_u^2+d_v^2)$. The augmented Zagreb index [31] was given by $AZI\left(G\right)={\sum }_{uv\in E\left(G\right)}{\left({\displaystyle \frac{d_u{d}_v}{d_u+d_v-2}}\right)}^3$. The Sombor index was defined in [32] as $SO\left(G\right)={\sum }_{uv\in E\left(G\right)}\sqrt{d_u^2+d_v^2}$. In 2024, Gutman [33] introduced the elliptic Sombor index: $ESO\left(G\right)={\sum }_{uv\in E\left(G\right)}(d_u+d_v)\sqrt{d_u^2+d_v^2}$.

Now we reproduce the definition of Shannon’s entropy [1]. Let $p=(p_1,p_2,\dots ,p_n)$ be a probability vector, namely, $0\le p_i\le 1$ and ${\sum }_{i=1}^n{p}_i=1$. The Shannon entropy of p is defined as

$$I_p=-\sum _{i=1}^n{p}_{i}log{p}_i.$$

In 2015, Chen et al. [4] introduced the definition of the entropy of edge-weighted graphs. For an edge-weighted graph G, the entropy of edge-weighted graph G is defined by

$$EN{T}_{\mu }\left(G\right)=-\sum _{uv\in E\left(G\right)}{\displaystyle \frac{\mu \left(uv\right)}{{\sum }_{uv\in E\left(G\right)}\mu \left(uv\right)}}log\left({\displaystyle \frac{\mu \left(uv\right)}{{\sum }_{uv\in E\left(G\right)}\mu \left(uv\right)}}\right)$$

(1)

where $\mu \left(uv\right)$ denotes the edge weight of edge $uv$. Note that

$$EN{T}_{\mu }\left(G\right)=log(\sum _{uv\in E\left(G\right)}\mu \left(uv\right))-{\displaystyle \frac{1}{{\sum }_{uv\in E\left(G\right)}\mu \left(uv\right)}}log\left(\prod _{uv\in E\left(G\right)}\mu {\left(uv\right)}^{\mu \left(uv\right)}\right).$$

(2)

Researchers in [34] have established the following entropies for edge-weighted graphs by assigning weights to edges that are equivalent to the main part of topological indices.

•

The first Zagreb entropy

If $\mu \left(uv\right)=(d_u+d_v)$, ${\sum }_{uv\in E\left(G\right)}\mu \left(uv\right)=M_1\left(G\right)$. The first Zagreb entropy is defined as follows [34]:

$$EN{T}_{M_1}\left(G\right)=log\left(M_1\left(G\right)\right)-{\displaystyle \frac{1}{M_1\left(G\right)}}log\left(\prod _{uv\in E\left(G\right)}{(d_u+d_v)}^{(d_u+d_v)}\right).$$

(3)

•

The second Zagreb entropy

If $\mu \left(uv\right)=d_u{d}_v$, ${\sum }_{uv\in E\left(G\right)}\mu \left(uv\right)=M_2\left(G\right)$. The second Zagreb entropy is [34]

$$EN{T}_{M_2}\left(G\right)=log\left(M_2\left(G\right)\right)-{\displaystyle \frac{1}{M_2\left(G\right)}}log\left(\prod _{uv\in E\left(G\right)}{\left(d_u{d}_v\right)}^{d_u{d}_v}\right).$$

(4)

•

The forgotten entropy

If $\mu \left(uv\right)=(d_u^2+d_v^2)$, ${\sum }_{uv\in E\left(G\right)}\mu \left(uv\right)=F\left(G\right)$. The forgotten entropy is [34]

$$EN{T}_F\left(G\right)=log\left(F\left(G\right)\right)-{\displaystyle \frac{1}{F\left(G\right)}}log\left(\prod _{uv\in E\left(G\right)}{(d_u^2+d_v^2)}^{(d_u^2+d_v^2)}\right).$$

(5)

•

The augmented Zagreb entropy

If $\mu \left(uv\right)={\left({\displaystyle \frac{d_u{d}_v}{d_u+d_v-2}}\right)}^3$, ${\sum }_{uv\in E\left(G\right)}\mu \left(uv\right)=AZI\left(G\right)$. The augmented Zagreb entropy is [34]

$$EN{T}_{AZI}\left(G\right)=log\left(AZI\left(G\right)\right)-{\displaystyle \frac{1}{AZI\left(G\right)}}log\left(\prod _{uv\in E\left(G\right)}{\left({\displaystyle \frac{d_u{d}_v}{d_u+d_v-2}}\right)}^{3{\left({\displaystyle \frac{d_u{d}_v}{d_u+d_v-2}}\right)}^3}\right).$$

(6)

•

The Sombor entropy

If $\mu \left(uv\right)=\sqrt{d_u^2+d_v^2}$, ${\sum }_{uv\in E\left(G\right)}\mu \left(uv\right)=SO\left(G\right)$. The Sombor entropy is

$$EN{T}_{SO}\left(G\right)=log\left(SO\left(G\right)\right)-{\displaystyle \frac{1}{SO\left(G\right)}}log\left(\prod _{uv\in E\left(G\right)}{\left(\sqrt{d_u^2+d_v^2}\right)}^{\sqrt{d_u^2+d_v^2}}\right).$$

(7)

•

The elliptic Sombor entropy

If $\mu \left(uv\right)=(d_u+d_v)\sqrt{d_u^2+d_v^2}$, ${\sum }_{uv\in E\left(G\right)}\mu \left(uv\right)=ESO\left(G\right)$. The elliptic Sombor entropy is

$$EN{T}_{ESO}\left(G\right)=log\left(ESO\left(G\right)\right)-{\displaystyle \frac{1}{ESO\left(G\right)}}log\left(\prod _{uv\in E\left(G\right)}{\left((d_u+d_v)\sqrt{d_u^2+d_v^2}\right)}^{(d_u+d_v)\sqrt{d_u^2+d_v^2}}\right).$$

(8)

In this paper, we employ the method of edge partition based on vertex degrees to derive analytical formulations for the first Zagreb entropy, the second Zagreb entropy, the forgotten entropy, the augmented Zagreb entropy, the Sombor entropy, and the elliptic Sombor entropy of the generalized Sierpiński graphs $S(M,t)$ and $S(N,t)$. Our results offer the possibility of better understanding their fundamental properties and potential applications in various fields such as chemistry, computer networks, and complex systems analysis.

2. Topological Indices and Entropies of the Generalized Sierpiński Graphs

In this section, we will investigate the topological indices and entropies of the generalized Sierpiński graphs with seed graphs M and N. The two graphs are depicted in Figure 1. In what follows, we derive the degree-based topological indices for the generalized Sierpiński graphs with seed graph M.

Theorem 1.

Let G be the generalized Sierpiński graph $S(M,t)$, where M is as shown in Figure 1, and $t\ge 2$. Then, the following hold:

1. 

$M_1\left(G\right)={\displaystyle \frac{35·4^t-62}{3}}$;

2. 

$M_2\left(G\right)=20·4^t-52$;

3. 

$F\left(G\right)={\displaystyle \frac{127·4^t-298}{3}}$;

4. 

$AZI\left(G\right)={\displaystyle \frac{\mathrm{230,968}}{\mathrm{10,125}}}·4^t-{\displaystyle \frac{\mathrm{9,767,803}}{\mathrm{162,000}}}$;

5. 

$SO\left(G\right)={\displaystyle \frac{2\sqrt{5}+4\sqrt{2}+15}{3}}{4}^t+{\displaystyle \frac{16\sqrt{5}+8\sqrt{2}-90}{3}}$;

6. 

$ESO\left(G\right)={\displaystyle \frac{12\sqrt{5}+32\sqrt{2}+105}{3}}{4}^t+{\displaystyle \frac{96\sqrt{5}-8\sqrt{2}-630}{3}}$.

Proof. 

We now introduce a notation: the number of edges whose end vertices have degrees k and $k^{\prime }$ will be denoted as $f_F(k,k^{\prime })$ for a given graph F.

For $S(M,2)$, the following values hold:

$$\begin{array}{cc}\hfill f_{S(M,2)}(2,4)& =8,\hfill \\ \hfill f_{S(M,2)}(3,3)& =4,\hfill \\ \hfill f_{S(M,2)}(3,4)& =10,\hfill \\ \hfill f_{S(M,2)}(4,4)& =3.\hfill \end{array}$$

Since $S(M,t)$ is constructed by replicating $S(M,t-1)$ four times and adding five edges, for $t>2$, $f_{S(M,t)}(a,b)$ can be computed from $f_{S(M,t-1)}(a,b)$. Specifically,

$$\begin{array}{cc}\hfill f_{S(M,t)}(2,4)& =4f_{S(M,t-1)}(2,4)-8,\hfill \\ \hfill f_{S(M,t)}(3,3)& =4,\hfill \\ \hfill f_{S(M,t)}(3,4)& =4f_{S(M,t-1)}(3,4)+18,\hfill \\ \hfill f_{S(M,t)}(4,4)& =4f_{S(M,t-1)}(3,4)+7.\hfill \end{array}$$

Hence, for $t\ge 2$,

$$\begin{array}{cc}\hfill f_{S(M,t)}(2,4)& ={\displaystyle \frac{4^t+8}{3}},\hfill \\ \hfill f_{S(M,t)}(3,3)& =4,\hfill \\ \hfill f_{S(M,t)}(3,4)& =4^t-6,\hfill \\ \hfill f_{S(M,t)}(4,4)& ={\displaystyle \frac{4^t-7}{3}}.\hfill \end{array}$$

Therefore, the edge distribution of G is detailed in Table 1.

Table 1. Edge distribution of G.

$({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$	(2,4)	(3,3)	(3,4)	(4,4)
$f_G(d_u,d_v)$	${\displaystyle \frac{4^t+8}{3}}$	4	$4^t-6$	${\displaystyle \frac{4^t-7}{3}}$

By applying the values from Table 1 and the definitions of six topological indices, we derive

$$\begin{array}{cc}\hfill M_1\left(G\right)=& \sum _{uv\in E\left(G\right)}(d_u+d_v)\hfill \\ \hfill =& \left({\displaystyle \frac{4^t+8}{3}}\right)\times 6+4\times 6+(4^t-6)\times 7+{\displaystyle \frac{4^t-7}{3}}\times 8\hfill \\ \hfill =& {\displaystyle \frac{35·4^t-62}{3}}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill M_2\left(G\right)=& \sum _{uv\in E\left(G\right)}{d}_u{d}_v\hfill \\ \hfill =& \left({\displaystyle \frac{4^t+8}{3}}\right)\times 8+4\times 9+(4^t-6)\times 12+{\displaystyle \frac{4^t-7}{3}}\times 16\hfill \\ \hfill =& 20·4^t-52.\hfill \end{array}$$

$$\begin{array}{cc}\hfill F\left(G\right)=& \sum _{uv\in E\left(G\right)}(d_u^2+d_v^2)\hfill \\ \hfill =& \left({\displaystyle \frac{4^t+8}{3}}\right)\times 20+4\times 18+(4^t-6)\times 25+{\displaystyle \frac{4^t-7}{3}}\times 32\hfill \\ \hfill =& {\displaystyle \frac{127·4^t-298}{3}}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill AZI\left(G\right)=& \sum _{uv\in E\left(G\right)}{\left({\displaystyle \frac{d_u{d}_v}{d_u+d_v-2}}\right)}^3\hfill \\ \hfill =& \left({\displaystyle \frac{4^t+8}{3}}\right)\times 8+4\times {\displaystyle \frac{729}{64}}+(4^t-6)\times {\displaystyle \frac{1728}{125}}+{\displaystyle \frac{4^t-7}{3}}\times {\displaystyle \frac{512}{27}}\hfill \\ \hfill =& {\displaystyle \frac{230968}{10125}}·4^t-{\displaystyle \frac{9767803}{162000}}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill SO\left(G\right)=& \sum _{uv\in E\left(G\right)}\sqrt{d_u^2+d_v^2}\hfill \\ \hfill =& \left({\displaystyle \frac{4^t+8}{3}}\right)\times 2\sqrt{5}+4\times 3\sqrt{2}+(4^t-6)\times 5+{\displaystyle \frac{4^t-7}{3}}\times 4\sqrt{2}\hfill \\ \hfill =& {\displaystyle \frac{2\sqrt{5}+4\sqrt{2}+15}{3}}{4}^t+{\displaystyle \frac{16\sqrt{5}+8\sqrt{2}-90}{3}}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill ESO\left(G\right)=& \sum _{uv\in E\left(G\right)}(d_u+d_v)\sqrt{d_u^2+d_v^2}\hfill \\ \hfill =& \left({\displaystyle \frac{4^t+8}{3}}\right)\times 12\sqrt{5}+4\times 18\sqrt{2}+(4^t-6)\times 35+{\displaystyle \frac{4^t-7}{3}}\times 32\sqrt{2}\hfill \\ \hfill =& {\displaystyle \frac{12\sqrt{5}+32\sqrt{2}+105}{3}}{4}^t+{\displaystyle \frac{96\sqrt{5}-8\sqrt{2}-630}{3}}.\hfill \end{array}$$

□

Next, the closed formulas of entropies of the generalized Sierpiński graphs $S(M,t)$ are given.

Theorem 2.

Let G be the generalized Sierpiński graph $S(M,t)$, where M is as shown in Figure 1, and $t\ge 2$. Then, the following hold:

1. 

$EN{T}_1\left(G\right)=log\left(M_1\left(G\right)\right)-{\displaystyle \frac{1}{M_1\left(G\right)}}log\left(2^{10\times 4^t-16)}\times 3^{2(4^t+20)}\times 7^{7(4^t-6)}\right)$;

2. 

$EN{T}_{M_2}\left(G\right)=log\left(M_2\left(G\right)\right)-{\displaystyle \frac{1}{M_2\left(G\right)}}log\left(2^{{\displaystyle \frac{1}{3}}(10\times 4^{t+2}-688)}\times 3^{3\left(4^{t+1}\right)}\right)$;

3. 

$EN{T}_F\left(G\right)=log\left(F\left(G\right)\right)-{\displaystyle \frac{1}{F\left(G\right)}}log\left(2^{{\displaystyle \frac{1}{3}}(200\times 4^t-584)}\times 3^{144}\times 5^{{\displaystyle \frac{1}{3}}(170\times 4^t-740)}\right)$;

4. 

$EN{T}_{AZI}\left(G\right)=\begin{array}{c}log\left(AZI\left(G\right)\right)-{\displaystyle \frac{1}{AZI\left(G\right)}}log(2^{{\displaystyle \frac{\mathrm{166,312}}{1125}}\times 4^t-{\displaystyle \frac{\mathrm{9,947,351}}{9000}}}\times 3^{{\displaystyle \frac{\mathrm{75,968}}{3375}}\times 4^t+{\displaystyle \frac{\mathrm{4,246,661}}{\mathrm{27,000}}}}\hfill \\ \times 5^{-{\displaystyle \frac{5184}{125}}(4^t-6)});\hfill \end{array}$

5. 

$EN{T}_{SO}\left(G\right)=\begin{array}{c}log\left(SO\left(G\right)\right)-{\displaystyle \frac{1}{SO\left(G\right)}}log(2^{{\displaystyle \frac{1}{3}}((2\sqrt{5}+10\sqrt{2})4^t+16\sqrt{5}-52\sqrt{2})}\times 3^{12\sqrt{2}}\hfill \\ \times 5^{{\displaystyle \frac{1}{3}}((\sqrt{5}+15)\times 4^t+8\sqrt{5}-90)});\hfill \end{array}$

6. 

$EN{T}_{ESO}\left(G\right)=\begin{array}{c}log\left(ESO\left(G\right)\right)-{\displaystyle \frac{1}{ESO\left(G\right)}}log(2^{{\displaystyle \frac{1}{3}}((24\sqrt{5}+176\sqrt{2})4^t+192\sqrt{5}-908\sqrt{2})}\hfill \\ \times 3^{4\sqrt{5}\times 4^t+32\sqrt{5}+144\sqrt{2}}\times 5^{(2\sqrt{5}+35)4^t+16\sqrt{5}-210}\times 7^{35(4^t-6)}).\hfill \end{array}$

Proof. 

To prove this, we refer to Table 1, which details the edge distribution of G. Then, by applying Equations (3)–(8) and Theorem 1, we conclude the following:

$$\begin{array}{cc}\hfill EN{T}_{M_1}\left(G\right)=& log\left(M_1\left(G\right)\right)-{\displaystyle \frac{1}{M_1\left(G\right)}}log\left(6^{6\times {\displaystyle \frac{4^t+8}{3}}}\times 6^{6\times 4}\times 7^{7\times (4^t-6)}\times 8^{8\times {\displaystyle \frac{4^t-7}{3}}}\right)\hfill \\ \hfill =& log\left(M_1\left(G\right)\right)-{\displaystyle \frac{1}{M_1\left(G\right)}}log\left(2^{10\times 4^t-16)}\times 3^{2(4^t+20)}\times 7^{7(4^t-6)}\right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_{M_2}\left(G\right)=& log\left(M_2\left(G\right)\right)-{\displaystyle \frac{1}{M_2\left(G\right)}}log\left(8^{8\times {\displaystyle \frac{4^t+8}{3}}}\times 9^{9\times 4}\times {12}^{12\times (4^t-6)}\times {16}^{16\times {\displaystyle \frac{4^t-7}{3}}}\right)\hfill \\ \hfill =& log\left(M_2\left(G\right)\right)-{\displaystyle \frac{1}{M_2\left(G\right)}}log\left(2^{{\displaystyle \frac{1}{3}}(10\times 4^{t+2}-688)}\times 3^{3\left(4^{t+1}\right)}\right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_F\left(G\right)=& log\left(F\left(G\right)\right)-{\displaystyle \frac{1}{F\left(G\right)}}log\left({20}^{20\times {\displaystyle \frac{4^t+8}{3}}}\times {18}^{18\times 4}\times {25}^{25\times (4^t-6)}\times {32}^{32\times {\displaystyle \frac{4^t-7}{3}}}\right)\hfill \\ \hfill =& log\left(F\left(G\right)\right)-{\displaystyle \frac{1}{F\left(G\right)}}log\left(2^{{\displaystyle \frac{1}{3}}(200\times 4^t-584)}\times 3^{144}\times 5^{{\displaystyle \frac{1}{3}}(170\times 4^t-740)}\right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_{AZI}\left(G\right)=& log\left(AZI\left(G\right)\right)-{\displaystyle \frac{1}{AZI\left(G\right)}}log(8^{8\times {\displaystyle \frac{4^t+8}{3}}}\times {{\displaystyle \frac{729}{64}}}^{{\displaystyle \frac{729}{64}}\times 4}\times {{\displaystyle \frac{1728}{125}}}^{{\displaystyle \frac{1728}{125}}\times (4^t-6)}\hfill \\ \hfill & \times {{\displaystyle \frac{512}{27}}}^{{\displaystyle \frac{512}{27}}\times {\displaystyle \frac{4^t-7}{3}}})\hfill \\ \hfill =& log\left(AZI\left(G\right)\right)-{\displaystyle \frac{1}{AZI\left(G\right)}}log(2^{{\displaystyle \frac{166,312}{1125}}\times 4^t-{\displaystyle \frac{9,947,351}{9000}}}\times 3^{{\displaystyle \frac{75,968}{3375}}\times 4^t+{\displaystyle \frac{4,246,661}{27,000}}}\hfill \\ \hfill & \times 5^{-{\displaystyle \frac{5184}{125}}(4^t-6)}).\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_{SO}\left(G\right)=& log\left(SO\left(G\right)\right)-{\displaystyle \frac{1}{SO\left(G\right)}}log(2{\sqrt{5}}^{2\sqrt{5}\times {\displaystyle \frac{4^t+8}{3}}}\times 3{\sqrt{2}}^{3\sqrt{2}\times 4}\hfill \\ \hfill & \times 5^{5(4^t-6)}\times 4{\sqrt{2}}^{4\sqrt{2}\times {\displaystyle \frac{4^t-7}{3}}})\hfill \\ \hfill =& log\left(SO\left(G\right)\right)-{\displaystyle \frac{1}{SO\left(G\right)}}log(2^{{\displaystyle \frac{1}{3}}((2\sqrt{5}+10\sqrt{2})4^t+16\sqrt{5}-52\sqrt{2})}\times 3^{12\sqrt{2}}\hfill \\ \hfill & \times 5^{{\displaystyle \frac{1}{3}}((\sqrt{5}+15)\times 4^t+8\sqrt{5}-90)}).\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_{ESO}\left(G\right)=& log\left(ESO\left(G\right)\right)-{\displaystyle \frac{1}{ESO\left(G\right)}}log(12{\sqrt{5}}^{12\sqrt{5}\times {\displaystyle \frac{4^t+8}{3}}}\times 18{\sqrt{2}}^{18\sqrt{2}\times 4}\hfill \\ \hfill & \times {35}^{35(4^t-6)}\times 32{\sqrt{2}}^{32\sqrt{2}\times {\displaystyle \frac{4^t-7}{3}}})\hfill \\ \hfill =& log\left(ESO\left(G\right)\right)-{\displaystyle \frac{1}{ESO\left(G\right)}}log(2^{{\displaystyle \frac{1}{3}}((24\sqrt{5}+176\sqrt{2})4^t+192\sqrt{5}-908\sqrt{2})}\hfill \\ \hfill & \times 3^{4\sqrt{5}\times 4^t+32\sqrt{5}+144\sqrt{2}}\times 5^{(2\sqrt{5}+35)4^t+16\sqrt{5}-210}\times 7^{35(4^t-6)}).\hfill \end{array}$$

□

Next, the closed formulas of the generalized Sierpiński graph with a house as its seed graph are given.

Theorem 3.

Let H be the generalized Sierpiński graph $S(N,t)$, where $t\ge 2$ and N is as shown in Figure 1. Then, the following hold:

1. 

$M_1\left(H\right)=48·5^{t-1}-18$;

2. 

$M_2\left(H\right)={\displaystyle \frac{1517·5^{t-2}-177}{4}}$;

3. 

$F\left(H\right)=162·5^{t-1}-84$;

4. 

$AZI\left(H\right)={\displaystyle \frac{\mathrm{127,687,649}}{\mathrm{288,000}}}·5^{t-2}-{\displaystyle \frac{\mathrm{44,461,087}}{\mathrm{864,000}}}$;

5. 

$SO\left(H\right)=({\displaystyle \frac{131\sqrt{2}}{4}}+5\sqrt{13}+16\sqrt{5}+{\displaystyle \frac{145}{2}})5^{t-2}+(\sqrt{13}+4\sqrt{5}-{\displaystyle \frac{7\sqrt{2}}{4}}-{\displaystyle \frac{45}{2}})$;

5. 

$ESO\left(H\right)=({\displaystyle \frac{445\sqrt{2}}{2}}+25\sqrt{13}+96\sqrt{5}+{\displaystyle \frac{1015}{2}})5^{t-2}+(5\sqrt{13}+24\sqrt{5}-{\displaystyle \frac{49\sqrt{2}}{2}}-{\displaystyle \frac{315}{2}})$.

Proof. 

For $S(N,2)$, the following values are established:

$$\begin{array}{cc}\hfill f_{S(N,2)}(2,2)& =1,\hfill \\ \hfill f_{S(N,2)}(2,3)& =6,\hfill \\ \hfill f_{S(N,2)}(2,4)& =10,\hfill \\ \hfill f_{S(N,2)}(3,3)& =7,\hfill \\ \hfill f_{S(N,2)}(3,4)& =10,\hfill \\ \hfill f_{S(N,2)}(4,4)& =2.\hfill \end{array}$$

Since $S(N,t)$ is constructed by replicating $S(N,t-1)$ five times and adding six edges, for $t>2$, $f_{S(N,t)}(a,b)$ can be computed from $f_{S(N,t-1)}(a,b)$. Specifically,

$$\begin{array}{cc}\hfill f_{S(N,t)}(2,2)& =5f_{S(N,t-1)}(2,2),\hfill \\ \hfill f_{S(N,t)}(2,3)& =5f_{S(N,t-1)}(2,3)-4,\hfill \\ \hfill f_{S(N,t)}(2,4)& =5f_{S(N,t-1)}(2,4)-8,\hfill \\ \hfill f_{S(N,t)}(3,3)& =5f_{S(N,t-1)}(3,3)-7,\hfill \\ \hfill f_{S(N,t)}(3,4)& =5f_{S(N,t-1)}(3,4)+18,\hfill \\ \hfill f_{S(N,t)}(4,4)& =5f_{S(N,t-1)}(4,4)+7.\hfill \end{array}$$

Hence, for $t\ge 2$,

$$\begin{array}{cc}\hfill f_{S(N,t)}(2,2)& =5^{t-2},\hfill \\ \hfill f_{S(N,t)}(2,3)& =5^{t-1}+1,\hfill \\ \hfill f_{S(N,t)}(2,4)& =8·5^{t-2}+2,\hfill \\ \hfill f_{S(N,t)}(3,3)& ={\displaystyle \frac{21·5^{t-2}+7}{4}},\hfill \\ \hfill f_{S(N,t)}(3,4)& ={\displaystyle \frac{29·5^{t-2}-9}{2}},\hfill \\ \hfill f_{S(N,t)}(4,4)& ={\displaystyle \frac{3·5^{t-1}-7}{4}}.\hfill \end{array}$$

Therefore, the edge distribution of H is detailed in Table 2.

Table 2. Edge distribution of H.

$({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$	(2,2)	(2,3)	(2,4)	(3,3)	(3,4)	(4,4)
$f_H(d_u,d_v)$	$5^{t-2}$	$5^{t-1}+1$	$8·5^{t-2}+2$	${\displaystyle \frac{21·5^{t-2}+7}{4}}$	${\displaystyle \frac{29·5^{t-2}-9}{2}}$	${\displaystyle \frac{3·5^{t-1}-7}{4}}$

Using the values provided in Table 2 along with the definitions of the six topological indices, we obtain the following:

$$\begin{array}{cc}\hfill M_1\left(H\right)=& \sum _{uv\in E\left(H\right)}(d_u+d_v)\hfill \\ \hfill =& 5^{t-2}\times 4+(5^{t-1}+1)\times 5+(8·5^{t-2}+2)\times 6+{\displaystyle \frac{21·5^{t-2}+7}{4}}\times 6\hfill \\ \hfill & +{\displaystyle \frac{29·5^{t-2}-9}{2}}\times 7+{\displaystyle \frac{3·5^{t-1}-7}{4}}\times 8\hfill \\ \hfill =& 48·5^{t-1}-18.\hfill \end{array}$$

$$\begin{array}{cc}\hfill M_2\left(H\right)=& \sum _{uv\in E\left(H\right)}{d}_u{d}_v\hfill \\ \hfill =& 5^{t-2}\times 4+(5^{t-1}+1)\times 6+(8·5^{t-2}+2)\times 8+{\displaystyle \frac{21·5^{t-2}+7}{4}}\times 9\hfill \\ \hfill & +{\displaystyle \frac{29·5^{t-2}-9}{2}}\times 12+{\displaystyle \frac{3·5^{t-1}-7}{4}}\times 16\hfill \\ \hfill =& {\displaystyle \frac{1517·5^{t-2}-177}{4}}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill F\left(H\right)=& \sum _{uv\in E\left(H\right)}(d_u^2+d_v^2)\hfill \\ \hfill =& 5^{t-2}\times 8+(5^{t-1}+1)\times 13+(8·5^{t-2}+2)\times 20+{\displaystyle \frac{21·5^{t-2}+7}{4}}\times 18\hfill \\ \hfill & +{\displaystyle \frac{29·5^{t-2}-9}{2}}\times 25+{\displaystyle \frac{3·5^{t-1}-7}{4}}\times 32\hfill \\ \hfill =& 162·5^{t-1}-84.\hfill \end{array}$$

$$\begin{array}{cc}\hfill AZI\left(H\right)=& \sum _{uv\in E\left(H\right)}{\left({\displaystyle \frac{d_u{d}_v}{d_u+d_v-2}}\right)}^3\hfill \\ \hfill =& 5^{t-2}\times 8+(5^{t-1}+1)\times 8+(8·5^{t-2}+2)\times 8+{\displaystyle \frac{21·5^{t-2}+7}{4}}\times {\displaystyle \frac{729}{64}}\hfill \\ \hfill & +{\displaystyle \frac{29·5^{t-2}-9}{2}}\times {\displaystyle \frac{1728}{125}}+{\displaystyle \frac{3·5^{t-1}-7}{4}}\times {\displaystyle \frac{512}{27}}\hfill \\ \hfill =& {\displaystyle \frac{\mathrm{127,687,649}}{\mathrm{288,000}}}·5^{t-2}-{\displaystyle \frac{\mathrm{44,461,087}}{\mathrm{864,000}}}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill SO\left(H\right)=& \sum _{uv\in E\left(H\right)}\sqrt{d_u^2+d_v^2}\hfill \\ \hfill =& 5^{t-2}\times 2\sqrt{2}+(5^{t-1}+1)\times \sqrt{13}+(8·5^{t-2}+2)\times 2\sqrt{5}+{\displaystyle \frac{21·5^{t-2}+7}{4}}\times 3\sqrt{2}\hfill \\ \hfill & +{\displaystyle \frac{29·5^{t-2}-9}{2}}\times 5+{\displaystyle \frac{3·5^{t-1}-7}{4}}\times 4\sqrt{2}\hfill \\ \hfill =& ({\displaystyle \frac{131\sqrt{2}}{4}}+5\sqrt{13}+16\sqrt{5}+{\displaystyle \frac{145}{2}})5^{t-2}+(\sqrt{13}+4\sqrt{5}-{\displaystyle \frac{7\sqrt{2}}{4}}-{\displaystyle \frac{45}{2}}).\hfill \end{array}$$

$$\begin{array}{cc}\hfill ESO\left(H\right)=& \sum _{uv\in E\left(H\right)}(d_u+d_v)\sqrt{d_u^2+d_v^2}\hfill \\ \hfill =& 5^{t-2}\times 8\sqrt{2}+(5^{t-1}+1)\times 5\sqrt{13}+(8·5^{t-2}+2)\times 12\sqrt{5}\hfill \\ \hfill & +{\displaystyle \frac{21·5^{t-2}+7}{4}}\times 18\sqrt{2}+{\displaystyle \frac{29·5^{t-2}-9}{2}}\times 35+{\displaystyle \frac{3·5^{t-1}-7}{4}}\times 32\sqrt{2}\hfill \\ \hfill =& ({\displaystyle \frac{445\sqrt{2}}{2}}+25\sqrt{13}+96\sqrt{5}+{\displaystyle \frac{1015}{2}})5^{t-2}+(5\sqrt{13}+24\sqrt{5}-{\displaystyle \frac{49\sqrt{2}}{2}}-{\displaystyle \frac{315}{2}}).\hfill \end{array}$$

□

For the next theorem, we will compute the closed formulas of entropies of the generalized Sierpiński graphs $S(N,t)$.

Theorem 4.

Let H be the generalized Sierpiński graph $S(N,t)$, where $t\ge 2$ and N is as shown in Figure 1. Then, the following hold:

1. 

$EN{T}_{M_1}\left(H\right)=\begin{array}{c}log\left(M_1\left(H\right)\right)-{\displaystyle \frac{1}{M_1\left(H\right)}}log(2^{{\displaystyle \frac{1}{2}}(71\times 5^{t-1}-39)}\times 3^{{\displaystyle \frac{1}{2}}(159\times 5^{t-2}+45)}\times 5^{5^t+5}\hfill \\ \times 7^{{\displaystyle \frac{7}{2}}(29\times 5^{t-2}-9)});\hfill \end{array}$

2. 

$EN{T}_{M_2}\left(H\right)=\begin{array}{c}log\left(M_2\left(H\right)\right)-{\displaystyle \frac{1}{M_2\left(H\right)}}log\left(2^{818\times 5^{t-2}-166}\times 3^{{\displaystyle \frac{1}{2}}(597\times 5^{t-2}-33)}\right);\hfill \end{array}$

3. 

$EN{T}_F\left(H\right)=\begin{array}{c}log\left(F\left(H\right)\right)-{\displaystyle \frac{1}{F\left(H\right)}}log(2^{{\displaystyle \frac{1}{2}}(2077\times 5^{t-2}-337)}\times 3^{189\times 5^{t-2}+63}\times 5^{885\times 5^{t-2}-185}\hfill \\ \times {13}^{13\times 5^{t-1}+13});\hfill \end{array}$

4. 

$EN{T}_{AZI}\left(H\right)=\begin{array}{c}log\left(AZI\left(H\right)\right)-{\displaystyle \frac{1}{AZI\left(H\right)}}log(2^{{\displaystyle \frac{1}{48,000}}(87,354,399\times 5^{t-2}-34,536,779)}\hfill \\ \times 3^{{\displaystyle \frac{1}{\mathrm{144,000}}}(\mathrm{107,541,411}\times 5^{t-2}+\mathrm{4,684,769})}\times 5^{-{\displaystyle \frac{2592}{125}}(29\times 5^{t-2}-9)});\hfill \end{array}$

5. 

$EN{T}_{SO}\left(H\right)=\begin{array}{c}log\left(SO\left(H\right)\right)-{\displaystyle \frac{1}{SO\left(H\right)}}log(2^{({\displaystyle \frac{387}{8}}\sqrt{2}+16\sqrt{5})\times 5^{t-2}+4\sqrt{5}-{\displaystyle \frac{119}{8}}\sqrt{2}}\hfill \\ \times 3^{{\displaystyle \frac{21}{4}}\sqrt{2}(3\times 5^{t-2}+1)}\times 5^{(8\sqrt{5}+{\displaystyle \frac{145}{2}})\times 5^{t-2}+2\sqrt{5}-{\displaystyle \frac{45}{2}}}\times {13}^{{\displaystyle \frac{\sqrt{13}}{2}}(5^{t-1}+1)});\hfill \end{array}$

6. 

$EN{T}_{ESO}\left(H\right)=\begin{array}{c}log\left(ESO\left(H\right)\right)-{\displaystyle \frac{1}{ESO\left(H\right)}}log(2^{({\displaystyle \frac{3319}{4}}\sqrt{2}+192\sqrt{5})\times 5^{t-2}+48\sqrt{5}-{\displaystyle \frac{1043}{4}}\sqrt{2}}\hfill \\ \times 3^{(189\sqrt{2}+96\sqrt{5})\times 5^{t-2}+24\sqrt{5}+63\sqrt{2}}\hfill \\ \times 5^{(25\sqrt{13}+48\sqrt{5}+{\displaystyle \frac{1015}{2}})\times 5^{t-2}+5\sqrt{13}+12\sqrt{5}-{\displaystyle \frac{315}{2}}}\hfill \\ \times 7^{{\displaystyle \frac{35}{2}}\times (29\times 5^{t-2}-9)}\times {13}^{{\displaystyle \frac{5\sqrt{13}}{2}}(5^{t-1}+1)}).\hfill \end{array}$

Proof. 

To prove this, we refer to Table 2, which provides the edge distribution of G. Then, by applying Equations (3)–(8) and Theorem 3, we can obtain the following expressions:

$$\begin{array}{cc}\hfill EN{T}_{M_1}\left(H\right)=& \begin{array}{c}log\left(M_1\left(H\right)\right)-{\displaystyle \frac{1}{M_1\left(H\right)}}log(4^{4\times 5^{t-2}}\times 5^{5\times (5^{t-1}+1)}\times 6^{6\times (8·5^{t-2}+2)}\hfill \\ \times 6^{6\times {\displaystyle \frac{21·5^{t-2}+7}{4}}}\times 7^{7\times {\displaystyle \frac{29·5^{t-2}-9}{2}}}\times 8^{8\times {\displaystyle \frac{3·5^{t-1}-7}{4}}})\hfill \end{array}\hfill \\ \hfill =& \begin{array}{c}log\left(M_1\left(H\right)\right)-{\displaystyle \frac{1}{M_1\left(H\right)}}log(2^{{\displaystyle \frac{1}{2}}(71\times 5^{t-1}-39)}\times 3^{{\displaystyle \frac{1}{2}}(159\times 5^{t-2}+45)}\times 5^{5^t+5}\hfill \\ \times 7^{{\displaystyle \frac{7}{2}}(29\times 5^{t-2}-9)}).\hfill \end{array}\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_{M_2}\left(H\right)=& \begin{array}{c}log\left(M_2\left(H\right)\right)-{\displaystyle \frac{1}{M_2\left(H\right)}}log(4^{4\times 5^{t-2}}\times 6^{6\times (5^{t-1}+1)}\times 8^{8\times (8·5^{t-2}+2)}\hfill \\ \times 9^{9\times {\displaystyle \frac{21·5^{t-2}+7}{4}}}\times {12}^{12\times {\displaystyle \frac{29·5^{t-2}-9}{2}}}\times {16}^{16\times {\displaystyle \frac{3·5^{t-1}-7}{4}}})\hfill \end{array}\hfill \\ \hfill =& \begin{array}{c}log\left(M_2\left(H\right)\right)-{\displaystyle \frac{1}{M_2\left(H\right)}}log\left(2^{818\times 5^{t-2}-166}\times 3^{{\displaystyle \frac{1}{2}}(597\times 5^{t-2}-33)}\right).\hfill \end{array}\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_F\left(H\right)=& \begin{array}{c}log\left(F\left(H\right)\right)-{\displaystyle \frac{1}{F\left(H\right)}}log(8^{8\times 5^{t-2}}\times {13}^{13\times (5^{t-1}+1)}\times {20}^{20\times (8·5^{t-2}+2)}\hfill \\ \times {18}^{18\times {\displaystyle \frac{21·5^{t-2}+7}{4}}}\times {25}^{25\times {\displaystyle \frac{29·5^{t-2}-9}{2}}}\times {32}^{32\times {\displaystyle \frac{3·5^{t-1}-7}{4}}})\hfill \end{array}\hfill \\ \hfill =& \begin{array}{c}log\left(F\left(H\right)\right)-{\displaystyle \frac{1}{F\left(H\right)}}log(2^{{\displaystyle \frac{1}{2}}(2077\times 5^{t-2}-337)}\times 3^{189\times 5^{t-2}+63}\times 5^{885\times 5^{t-2}-185}\hfill \\ \times {13}^{13\times 5^{t-1}+13}).\hfill \end{array}\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_{AZI}\left(H\right)=& \begin{array}{c}log\left(AZI\left(H\right)\right)-{\displaystyle \frac{1}{AZI\left(H\right)}}log(8^{8\times 5^{t-2}}\times 8^{8\times (5^{t-1}+1)}\times 8^{8\times (8·5^{t-2}+2)}\hfill \\ \times {{\displaystyle \frac{729}{64}}}^{{\displaystyle \frac{729}{64}}\times {\displaystyle \frac{21·5^{t-2}+7}{4}}}\times {{\displaystyle \frac{1728}{125}}}^{{\displaystyle \frac{1728}{125}}\times {\displaystyle \frac{29·5^{t-2}-9}{2}}}\times {{\displaystyle \frac{512}{27}}}^{{\displaystyle \frac{512}{27}}\times {\displaystyle \frac{3·5^{t-1}-7}{4}}})\hfill \end{array}\hfill \\ \hfill =& \begin{array}{c}log\left(AZI\left(H\right)\right)-{\displaystyle \frac{1}{AZI\left(H\right)}}log(2^{{\displaystyle \frac{1}{\mathrm{48,000}}}(\mathrm{87,354,399}\times 5^{t-2}-\mathrm{34,536,779})}\hfill \\ \times 3^{{\displaystyle \frac{1}{\mathrm{144,000}}}(\mathrm{107,541,411}\times 5^{t-2}+\mathrm{4,684,769})}\times 5^{-{\displaystyle \frac{2592}{125}}(29\times 5^{t-2}-9)}).\hfill \end{array}\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_{SO}\left(H\right)=& \begin{array}{c}log\left(SO\left(H\right)\right)-{\displaystyle \frac{1}{SO\left(H\right)}}log({\left(2\sqrt{2}\right)}^{2\sqrt{2}\times 5^{t-2}}\times {\sqrt{13}}^{\sqrt{13}\times (5^{t-1}+1)}\hfill \\ \times {\left(2\sqrt{5}\right)}^{2\sqrt{5}\times (8·5^{t-2}+2)}\times {\left(3\sqrt{2}\right)}^{3\sqrt{2}\times {\displaystyle \frac{21·5^{t-2}+7}{4}}}\hfill \\ \times 5^{5\times {\displaystyle \frac{29·5^{t-2}-9}{2}}}\times {\left(4\sqrt{2}\right)}^{4\sqrt{2}\times {\displaystyle \frac{3·5^{t-1}-7}{4}}})\hfill \end{array}\hfill \\ \hfill =& \begin{array}{c}log\left(SO\left(H\right)\right)-{\displaystyle \frac{1}{SO\left(H\right)}}log(2^{({\displaystyle \frac{387}{8}}\sqrt{2}+16\sqrt{5})\times 5^{t-2}+4\sqrt{5}-{\displaystyle \frac{119}{8}}\sqrt{2}}\hfill \\ \times 3^{{\displaystyle \frac{21}{4}}\sqrt{2}(3\times 5^{t-2}+1)}\times 5^{(8\sqrt{5}+{\displaystyle \frac{145}{2}})\times 5^{t-2}+2\sqrt{5}-{\displaystyle \frac{45}{2}}}\times {13}^{{\displaystyle \frac{\sqrt{13}}{2}}(5^{t-1}+1)}).\hfill \end{array}\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_{ESO}\left(H\right)=& \begin{array}{c}log\left(ESO\left(H\right)\right)-{\displaystyle \frac{1}{ESO\left(H\right)}}log({\left(8\sqrt{2}\right)}^{8\sqrt{2}\times 5^{t-2}}\times {\left(5\sqrt{13}\right)}^{5\sqrt{13}\times (5^{t-1}+1)}\hfill \\ \times {\left(12\sqrt{5}\right)}^{12\sqrt{5}\times (8·5^{t-2}+2)}\times {\left(18\sqrt{2}\right)}^{18\sqrt{2}\times {\displaystyle \frac{21·5^{t-2}+7}{4}}}\hfill \\ \times {35}^{35\times {\displaystyle \frac{29·5^{t-2}-9}{2}}}\times {\left(32\sqrt{2}\right)}^{32\sqrt{2}\times {\displaystyle \frac{3·5^{t-1}-7}{4}}})\hfill \end{array}\hfill \\ \hfill =& \begin{array}{c}log\left(ESO\left(H\right)\right)-{\displaystyle \frac{1}{ESO\left(H\right)}}log(2^{({\displaystyle \frac{3319}{4}}\sqrt{2}+192\sqrt{5})\times 5^{t-2}+48\sqrt{5}-{\displaystyle \frac{1043}{4}}\sqrt{2}}\hfill \\ \times 3^{(189\sqrt{2}+96\sqrt{5})\times 5^{t-2}+24\sqrt{5}+63\sqrt{2}}\hfill \\ \times 5^{(25\sqrt{13}+48\sqrt{5}+{\displaystyle \frac{1015}{2}})\times 5^{t-2}+5\sqrt{13}+12\sqrt{5}-{\displaystyle \frac{315}{2}}}\hfill \\ \times 7^{{\displaystyle \frac{35}{2}}\times (29\times 5^{t-2}-9)}\times {13}^{{\displaystyle \frac{5\sqrt{13}}{2}}(5^{t-1}+1)}).\hfill \end{array}\hfill \end{array}$$

□

We compare the entropies related to the first Zagreb entropy, the second Zagreb entropy, the forgotten entropy, the augmented Zagreb entropy, the Sombor entropy, and the elliptic Sombor entropy for $S(M,t)$ and $S(N,t)$ numerically and graphically in Table 3 and Table 4 and Figure 4, respectively. These entropies are systematically evaluated to provide comprehensive insights into the topological properties of the studied graphs.

Table 3. Entropy values of $EN{T}_f$ for the generalized Sierpiński graphs $S(M,t)$.

f	$\mathit{t}=2$	$\mathit{t}=3$	$\mathit{t}=4$	$\mathit{t}=5$	$\mathit{t}=6$	$\mathit{t}=7$	$\mathit{t}=8$	$\mathit{t}=9$	$\mathit{t}=10$
$M_1$	3.2136	4.6494	6.0479	7.4372	8.8242	10.2107	11.5971	12.9834	14.3697
$M_2$	3.19	4.6288	6.0288	7.4185	8.8057	10.1922	11.5785	12.9648	14.3511
F	3.2024	4.6408	6.0404	7.43	8.8171	10.2036	11.59	12.9763	14.3626
$AZI$	3.1777	4.6175	6.0177	7.4075	8.7947	10.1812	11.5676	12.9539	14.3402
$SO$	3.2148	4.6507	6.0492	7.4385	8.8256	10.212	11.5984	12.9847	14.371
$ESO$	3.2001	4.6384	6.038	7.4276	8.8148	10.2013	11.5876	12.9739	14.3602

Table 4. Entropy values of $EN{T}_f$ for the generalized Sierpiński graphs $S(N,t)$.

f	$\mathit{t}=2$	$\mathit{t}=3$	$\mathit{t}=4$	$\mathit{t}=5$	$\mathit{t}=6$	$\mathit{t}=7$	$\mathit{t}=8$	$\mathit{t}=9$	$\mathit{t}=10$
$M_1$	3.5735	5.2149	6.8308	8.4415	10.0512	11.6607	13.2701	14.8795	16.489
$M_2$	3.5403	5.1798	6.7957	8.4064	10.0161	11.6256	13.2351	14.8445	16.454
F	3.5482	5.189	6.805	8.4157	10.0254	11.6349	13.2443	14.8538	16.4632
$AZI$	3.5432	5.1829	6.7988	8.4096	10.0193	11.6287	13.2382	14.8476	16.4571
$SO$	3.574	5.2157	6.8316	8.4423	10.052	11.6615	13.2709	14.8804	16.4898
$ESO$	3.547	5.1874	6.8033	8.414	10.0237	11.6332	13.2426	14.8521	16.4615

Figure 4. Graphical representation of entropy values of $S(M,t)$ and $S(N,t)$ for six degree-based topological indices.

From the analysis of Table 3 and Table 4 and Figure 4, it is evident that as t increases in graphs $S(M,t)$ and $S(N,t)$, all entropies display a significant increasing trend. Moreover, for graph $S(M,t)$,

$$\begin{array}{c}EN{T}_{SO}\left(S(M,t)\right)>EN{T}_{M_1}\left(S(M,t)\right)>EN{T}_F\left(S(M,t)\right)\hfill \\ \hfill >EN{T}_{ESO}\left(S(M,t)\right)>EN{T}_{M_2}\left(S(M,t)\right)>EN{T}_{AZI}\left(S(M,t)\right);\end{array}$$

for graph $S(N,t)$,

$$\begin{array}{c}EN{T}_{SO}\left(S(N,t)\right)>EN{T}_{M_1}\left(S(N,t)\right)>EN{T}_F\left(S(N,t)\right)\hfill \\ \hfill >EN{T}_{ESO}\left(S(N,t)\right)>EN{T}_{AZI}\left(S(N,t)\right)>EN{T}_{M_2}\left(S(N,t)\right).\end{array}$$

3. Conclusions

This paper investigates two types of generalized Sierpiński graphs based on different seed graphs. Utilizing various entropy-based methods and topological indices, including the first Zagreb index, the second Zagreb index, the forgotten index, the augmented Zagreb index, the Sombor index, and the elliptic Sombor index, this study provides a comprehensive understanding of the complex structural characteristics of generalized Sierpiński graphs. The integration of entropy with generalized Sierpiński graphs offers novel perspectives for addressing complex challenges in molecular chemistry, computer networks, complex systems analysis, and related fields. Future research can expand the applications of these networks in various fields based on these findings.