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Article

Entropiesand Degree-Based Topological Indices of Generalized Sierpiński Graphs

1
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
2
School of Electronics and Information Engineering, Anhui Jianzhu University, Hefei 230601, China
3
School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China
*
Author to whom correspondence should be addressed.
Fractal Fract. 2025, 9(3), 190; https://doi.org/10.3390/fractalfract9030190
Submission received: 20 February 2025 / Revised: 14 March 2025 / Accepted: 17 March 2025 / Published: 19 March 2025
(This article belongs to the Special Issue Fractal Functions: Theoretical Research and Application Analysis)

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 ( G ) and E ( G ) 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 ( G ) = { 1 , 2 , , n } , the graph S ( G , t ) , referred to as the generalized Sierpiński graph of G [22], is constructed as follows:
V ( S ( G , t ) ) = { x 1 x 2 x t x i V ( G ) for 1 i t } .
E ( S ( G , t ) ) : For two vertices x = x 1 x 2 x t and y = y 1 y 2 y t in V ( S ( G , t ) ) , { x , y } E ( S ( G , t ) ) if there exists p { 1 , 2 , , t } such that
(i)
For all q < p , x q = y q ;
(ii)
x p y p and { x p , y p } E ( G ) ;
(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 x i 1 such that x can be formed by concatenating w with the alternating sequence x i y i y i , and similarly for y with w followed by y i x i 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 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 g h h h and h g g 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 ( N , 1 ) , S ( N , 2 ) , S ( N , 3 ) are displayed in Figure 3.
Next, we introduce the definitions of six degree-based topological indices. The first and second Zagreb indices [29] were defined as M 1 ( G ) = u v E ( G ) ( d u + d v ) and M 2 ( G ) = u v E ( G ) d u d v . The forgotten index [30] was given by F ( G ) = u v E ( G ) ( d u 2 + d v 2 ) . The augmented Zagreb index [31] was given by A Z I ( G ) = u v E ( G ) d u d v d u + d v 2 3 . The Sombor index was defined in [32] as S O ( G ) = u v E ( G ) d u 2 + d v 2 . In 2024, Gutman [33] introduced the elliptic Sombor index: E S O ( G ) = u v E ( G ) ( d u + d v ) d u 2 + d v 2 .
Now we reproduce the definition of Shannon’s entropy [1]. Let p = ( p 1 , p 2 , , p n ) be a probability vector, namely, 0 p i 1 and i = 1 n p i = 1 . The Shannon entropy of p is defined as
I p = 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
E N T μ ( G ) = u v E ( G ) μ ( u v ) u v E ( G ) μ ( u v ) log μ ( u v ) u v E ( G ) μ ( u v )
where μ ( u v ) denotes the edge weight of edge u v . Note that
E N T μ ( G ) = log ( u v E ( G ) μ ( u v ) ) 1 u v E ( G ) μ ( u v ) log u v E ( G ) μ ( u v ) μ ( u v ) .
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 μ ( u v ) = ( d u + d v ) , u v E ( G ) μ ( u v ) = M 1 ( G ) . The first Zagreb entropy is defined as follows [34]:
E N T M 1 ( G ) = log ( M 1 ( G ) ) 1 M 1 ( G ) log u v E ( G ) ( d u + d v ) ( d u + d v ) .
The second Zagreb entropy
If μ ( u v ) = d u d v , u v E ( G ) μ ( u v ) = M 2 ( G ) . The second Zagreb entropy is [34]
E N T M 2 ( G ) = log ( M 2 ( G ) ) 1 M 2 ( G ) log u v E ( G ) d u d v d u d v .
The forgotten entropy
If μ ( u v ) = ( d u 2 + d v 2 ) , u v E ( G ) μ ( u v ) = F ( G ) . The forgotten entropy is [34]
E N T F ( G ) = log ( F ( G ) ) 1 F ( G ) log u v E ( G ) ( d u 2 + d v 2 ) ( d u 2 + d v 2 ) .
The augmented Zagreb entropy
If μ ( u v ) = d u d v d u + d v 2 3 , u v E ( G ) μ ( u v ) = A Z I ( G ) . The augmented Zagreb entropy is [34]
E N T A Z I ( G ) = log ( A Z I ( G ) ) 1 A Z I ( G ) log u v E ( G ) d u d v d u + d v 2 3 d u d v d u + d v 2 3 .
The Sombor entropy
If μ ( u v ) = d u 2 + d v 2 , u v E ( G ) μ ( u v ) = S O ( G ) . The Sombor entropy is
E N T S O ( G ) = log ( S O ( G ) ) 1 S O ( G ) log u v E ( G ) d u 2 + d v 2 d u 2 + d v 2 .
The elliptic Sombor entropy
If μ ( u v ) = ( d u + d v ) d u 2 + d v 2 , u v E ( G ) μ ( u v ) = E S O ( G ) . The elliptic Sombor entropy is
E N T E S O ( G ) = log ( E S O ( G ) ) 1 E S O ( G ) log u v E ( G ) ( d u + d v ) d u 2 + d v 2 ( d u + d v ) d u 2 + d v 2 .
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 2 . Then, the following hold:
1. 
M 1 ( G ) = 35 · 4 t 62 3 ;
2. 
M 2 ( G ) = 20 · 4 t 52 ;
3. 
F ( G ) = 127 · 4 t 298 3 ;
4. 
A Z I ( G ) = 230,968 10,125 · 4 t 9,767,803 162,000 ;
5. 
S O ( G ) = 2 5 + 4 2 + 15 3 4 t + 16 5 + 8 2 90 3 ;
6. 
E S O ( G ) = 12 5 + 32 2 + 105 3 4 t + 96 5 8 2 630 3 .
Proof. 
We now introduce a notation: the number of edges whose end vertices have degrees k and k will be denoted as f F ( k , k ) for a given graph F.
For S ( M , 2 ) , the following values hold:
f S ( M , 2 ) ( 2 , 4 ) = 8 , f S ( M , 2 ) ( 3 , 3 ) = 4 , f S ( M , 2 ) ( 3 , 4 ) = 10 , f S ( M , 2 ) ( 4 , 4 ) = 3 .
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,
f S ( M , t ) ( 2 , 4 ) = 4 f S ( M , t 1 ) ( 2 , 4 ) 8 , f S ( M , t ) ( 3 , 3 ) = 4 , f S ( M , t ) ( 3 , 4 ) = 4 f S ( M , t 1 ) ( 3 , 4 ) + 18 , f S ( M , t ) ( 4 , 4 ) = 4 f S ( M , t 1 ) ( 3 , 4 ) + 7 .
Hence, for t 2 ,
f S ( M , t ) ( 2 , 4 ) = 4 t + 8 3 , f S ( M , t ) ( 3 , 3 ) = 4 , f S ( M , t ) ( 3 , 4 ) = 4 t 6 , f S ( M , t ) ( 4 , 4 ) = 4 t 7 3 .
Therefore, the edge distribution of G is detailed in Table 1.
By applying the values from Table 1 and the definitions of six topological indices, we derive
M 1 ( G ) = u v E ( G ) ( d u + d v ) = ( 4 t + 8 3 ) × 6 + 4 × 6 + ( 4 t 6 ) × 7 + 4 t 7 3 × 8 = 35 · 4 t 62 3 .
M 2 ( G ) = u v E ( G ) d u d v = ( 4 t + 8 3 ) × 8 + 4 × 9 + ( 4 t 6 ) × 12 + 4 t 7 3 × 16 = 20 · 4 t 52 .
F ( G ) = u v E ( G ) ( d u 2 + d v 2 ) = ( 4 t + 8 3 ) × 20 + 4 × 18 + ( 4 t 6 ) × 25 + 4 t 7 3 × 32 = 127 · 4 t 298 3 .
A Z I ( G ) = u v E ( G ) d u d v d u + d v 2 3 = ( 4 t + 8 3 ) × 8 + 4 × 729 64 + ( 4 t 6 ) × 1728 125 + 4 t 7 3 × 512 27 = 230968 10125 · 4 t 9767803 162000 .
S O ( G ) = u v E ( G ) d u 2 + d v 2 = ( 4 t + 8 3 ) × 2 5 + 4 × 3 2 + ( 4 t 6 ) × 5 + 4 t 7 3 × 4 2 = 2 5 + 4 2 + 15 3 4 t + 16 5 + 8 2 90 3 .
E S O ( G ) = u v E ( G ) ( d u + d v ) d u 2 + d v 2 = ( 4 t + 8 3 ) × 12 5 + 4 × 18 2 + ( 4 t 6 ) × 35 + 4 t 7 3 × 32 2 = 12 5 + 32 2 + 105 3 4 t + 96 5 8 2 630 3 .
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 2 . Then, the following hold:
1. 
E N T 1 ( G ) = log ( M 1 ( G ) ) 1 M 1 ( G ) log 2 10 × 4 t 16 ) × 3 2 ( 4 t + 20 ) × 7 7 ( 4 t 6 ) ;
2. 
E N T M 2 ( G ) = log ( M 2 ( G ) ) 1 M 2 ( G ) log 2 1 3 ( 10 × 4 t + 2 688 ) × 3 3 ( 4 t + 1 ) ;
3. 
E N T F ( G ) = log ( F ( G ) ) 1 F ( G ) log 2 1 3 ( 200 × 4 t 584 ) × 3 144 × 5 1 3 ( 170 × 4 t 740 ) ;
4. 
E N T A Z I ( G ) = log ( A Z I ( G ) ) 1 A Z I ( G ) log ( 2 166,312 1125 × 4 t 9,947,351 9000 × 3 75,968 3375 × 4 t + 4,246,661 27,000 × 5 5184 125 ( 4 t 6 ) ) ;
5. 
E N T S O ( G ) = log ( S O ( G ) ) 1 S O ( G ) log ( 2 1 3 ( ( 2 5 + 10 2 ) 4 t + 16 5 52 2 ) × 3 12 2 × 5 1 3 ( ( 5 + 15 ) × 4 t + 8 5 90 ) ) ;
6. 
E N T E S O ( G ) = log ( E S O ( G ) ) 1 E S O ( G ) log ( 2 1 3 ( ( 24 5 + 176 2 ) 4 t + 192 5 908 2 ) × 3 4 5 × 4 t + 32 5 + 144 2 × 5 ( 2 5 + 35 ) 4 t + 16 5 210 × 7 35 ( 4 t 6 ) ) .
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:
E N T M 1 ( G ) = log ( M 1 ( G ) ) 1 M 1 ( G ) log 6 6 × 4 t + 8 3 × 6 6 × 4 × 7 7 × ( 4 t 6 ) × 8 8 × 4 t 7 3 = log ( M 1 ( G ) ) 1 M 1 ( G ) log 2 10 × 4 t 16 ) × 3 2 ( 4 t + 20 ) × 7 7 ( 4 t 6 ) .
E N T M 2 ( G ) = log ( M 2 ( G ) ) 1 M 2 ( G ) log 8 8 × 4 t + 8 3 × 9 9 × 4 × 12 12 × ( 4 t 6 ) × 16 16 × 4 t 7 3 = log ( M 2 ( G ) ) 1 M 2 ( G ) log 2 1 3 ( 10 × 4 t + 2 688 ) × 3 3 ( 4 t + 1 ) .
E N T F ( G ) = log ( F ( G ) ) 1 F ( G ) log 20 20 × 4 t + 8 3 × 18 18 × 4 × 25 25 × ( 4 t 6 ) × 32 32 × 4 t 7 3 = log ( F ( G ) ) 1 F ( G ) log 2 1 3 ( 200 × 4 t 584 ) × 3 144 × 5 1 3 ( 170 × 4 t 740 ) .
E N T A Z I ( G ) = log ( A Z I ( G ) ) 1 A Z I ( G ) log ( 8 8 × 4 t + 8 3 × 729 64 729 64 × 4 × 1728 125 1728 125 × ( 4 t 6 ) × 512 27 512 27 × 4 t 7 3 ) = log ( A Z I ( G ) ) 1 A Z I ( G ) log ( 2 166 , 312 1125 × 4 t 9 , 947 , 351 9000 × 3 75 , 968 3375 × 4 t + 4 , 246 , 661 27 , 000 × 5 5184 125 ( 4 t 6 ) ) .
E N T S O ( G ) = log ( S O ( G ) ) 1 S O ( G ) log ( 2 5 2 5 × 4 t + 8 3 × 3 2 3 2 × 4 × 5 5 ( 4 t 6 ) × 4 2 4 2 × 4 t 7 3 ) = log ( S O ( G ) ) 1 S O ( G ) log ( 2 1 3 ( ( 2 5 + 10 2 ) 4 t + 16 5 52 2 ) × 3 12 2 × 5 1 3 ( ( 5 + 15 ) × 4 t + 8 5 90 ) ) .
E N T E S O ( G ) = log ( E S O ( G ) ) 1 E S O ( G ) log ( 12 5 12 5 × 4 t + 8 3 × 18 2 18 2 × 4 × 35 35 ( 4 t 6 ) × 32 2 32 2 × 4 t 7 3 ) = log ( E S O ( G ) ) 1 E S O ( G ) log ( 2 1 3 ( ( 24 5 + 176 2 ) 4 t + 192 5 908 2 ) × 3 4 5 × 4 t + 32 5 + 144 2 × 5 ( 2 5 + 35 ) 4 t + 16 5 210 × 7 35 ( 4 t 6 ) ) .
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 2 and N is as shown in Figure 1. Then, the following hold:
1. 
M 1 ( H ) = 48 · 5 t 1 18 ;
2. 
M 2 ( H ) = 1517 · 5 t 2 177 4 ;
3. 
F ( H ) = 162 · 5 t 1 84 ;
4. 
A Z I ( H ) = 127,687,649 288,000 · 5 t 2 44,461,087 864,000 ;
5. 
S O ( H ) = ( 131 2 4 + 5 13 + 16 5 + 145 2 ) 5 t 2 + ( 13 + 4 5 7 2 4 45 2 ) ;
5. 
E S O ( H ) = ( 445 2 2 + 25 13 + 96 5 + 1015 2 ) 5 t 2 + ( 5 13 + 24 5 49 2 2 315 2 ) .
Proof. 
For S ( N , 2 ) , the following values are established:
f S ( N , 2 ) ( 2 , 2 ) = 1 , f S ( N , 2 ) ( 2 , 3 ) = 6 , f S ( N , 2 ) ( 2 , 4 ) = 10 , f S ( N , 2 ) ( 3 , 3 ) = 7 , f S ( N , 2 ) ( 3 , 4 ) = 10 , f S ( N , 2 ) ( 4 , 4 ) = 2 .
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,
f S ( N , t ) ( 2 , 2 ) = 5 f S ( N , t 1 ) ( 2 , 2 ) , f S ( N , t ) ( 2 , 3 ) = 5 f S ( N , t 1 ) ( 2 , 3 ) 4 , f S ( N , t ) ( 2 , 4 ) = 5 f S ( N , t 1 ) ( 2 , 4 ) 8 , f S ( N , t ) ( 3 , 3 ) = 5 f S ( N , t 1 ) ( 3 , 3 ) 7 , f S ( N , t ) ( 3 , 4 ) = 5 f S ( N , t 1 ) ( 3 , 4 ) + 18 , f S ( N , t ) ( 4 , 4 ) = 5 f S ( N , t 1 ) ( 4 , 4 ) + 7 .
Hence, for t 2 ,
f S ( N , t ) ( 2 , 2 ) = 5 t 2 , f S ( N , t ) ( 2 , 3 ) = 5 t 1 + 1 , f S ( N , t ) ( 2 , 4 ) = 8 · 5 t 2 + 2 , f S ( N , t ) ( 3 , 3 ) = 21 · 5 t 2 + 7 4 , f S ( N , t ) ( 3 , 4 ) = 29 · 5 t 2 9 2 , f S ( N , t ) ( 4 , 4 ) = 3 · 5 t 1 7 4 .
Therefore, the edge distribution of H is detailed in Table 2.
Using the values provided in Table 2 along with the definitions of the six topological indices, we obtain the following:
M 1 ( H ) = u v E ( H ) ( d u + d v ) = 5 t 2 × 4 + ( 5 t 1 + 1 ) × 5 + ( 8 · 5 t 2 + 2 ) × 6 + 21 · 5 t 2 + 7 4 × 6 + 29 · 5 t 2 9 2 × 7 + 3 · 5 t 1 7 4 × 8 = 48 · 5 t 1 18 .
M 2 ( H ) = u v E ( H ) d u d v = 5 t 2 × 4 + ( 5 t 1 + 1 ) × 6 + ( 8 · 5 t 2 + 2 ) × 8 + 21 · 5 t 2 + 7 4 × 9 + 29 · 5 t 2 9 2 × 12 + 3 · 5 t 1 7 4 × 16 = 1517 · 5 t 2 177 4 .
F ( H ) = u v E ( H ) ( d u 2 + d v 2 ) = 5 t 2 × 8 + ( 5 t 1 + 1 ) × 13 + ( 8 · 5 t 2 + 2 ) × 20 + 21 · 5 t 2 + 7 4 × 18 + 29 · 5 t 2 9 2 × 25 + 3 · 5 t 1 7 4 × 32 = 162 · 5 t 1 84 .
A Z I ( H ) = u v E ( H ) d u d v d u + d v 2 3 = 5 t 2 × 8 + ( 5 t 1 + 1 ) × 8 + ( 8 · 5 t 2 + 2 ) × 8 + 21 · 5 t 2 + 7 4 × 729 64 + 29 · 5 t 2 9 2 × 1728 125 + 3 · 5 t 1 7 4 × 512 27 = 127,687,649 288,000 · 5 t 2 44,461,087 864,000 .
S O ( H ) = u v E ( H ) d u 2 + d v 2 = 5 t 2 × 2 2 + ( 5 t 1 + 1 ) × 13 + ( 8 · 5 t 2 + 2 ) × 2 5 + 21 · 5 t 2 + 7 4 × 3 2 + 29 · 5 t 2 9 2 × 5 + 3 · 5 t 1 7 4 × 4 2 = ( 131 2 4 + 5 13 + 16 5 + 145 2 ) 5 t 2 + ( 13 + 4 5 7 2 4 45 2 ) .
E S O ( H ) = u v E ( H ) ( d u + d v ) d u 2 + d v 2 = 5 t 2 × 8 2 + ( 5 t 1 + 1 ) × 5 13 + ( 8 · 5 t 2 + 2 ) × 12 5 + 21 · 5 t 2 + 7 4 × 18 2 + 29 · 5 t 2 9 2 × 35 + 3 · 5 t 1 7 4 × 32 2 = ( 445 2 2 + 25 13 + 96 5 + 1015 2 ) 5 t 2 + ( 5 13 + 24 5 49 2 2 315 2 ) .
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 2 and N is as shown in Figure 1. Then, the following hold:
1. 
E N T M 1 ( H ) = log ( M 1 ( H ) ) 1 M 1 ( H ) log ( 2 1 2 ( 71 × 5 t 1 39 ) × 3 1 2 ( 159 × 5 t 2 + 45 ) × 5 5 t + 5 × 7 7 2 ( 29 × 5 t 2 9 ) ) ;
2. 
E N T M 2 ( H ) = log ( M 2 ( H ) ) 1 M 2 ( H ) log 2 818 × 5 t 2 166 × 3 1 2 ( 597 × 5 t 2 33 ) ;
3. 
E N T F ( H ) = log ( F ( H ) ) 1 F ( H ) log ( 2 1 2 ( 2077 × 5 t 2 337 ) × 3 189 × 5 t 2 + 63 × 5 885 × 5 t 2 185 × 13 13 × 5 t 1 + 13 ) ;
4. 
E N T A Z I ( H ) = log ( A Z I ( H ) ) 1 A Z I ( H ) log ( 2 1 48 , 000 ( 87 , 354 , 399 × 5 t 2 34 , 536 , 779 ) × 3 1 144,000 ( 107,541,411 × 5 t 2 + 4,684,769 ) × 5 2592 125 ( 29 × 5 t 2 9 ) ) ;
5. 
E N T S O ( H ) = log ( S O ( H ) ) 1 S O ( H ) log ( 2 ( 387 8 2 + 16 5 ) × 5 t 2 + 4 5 119 8 2 × 3 21 4 2 ( 3 × 5 t 2 + 1 ) × 5 ( 8 5 + 145 2 ) × 5 t 2 + 2 5 45 2 × 13 13 2 ( 5 t 1 + 1 ) ) ;
6. 
E N T E S O ( H ) = log ( E S O ( H ) ) 1 E S O ( H ) log ( 2 ( 3319 4 2 + 192 5 ) × 5 t 2 + 48 5 1043 4 2 × 3 ( 189 2 + 96 5 ) × 5 t 2 + 24 5 + 63 2 × 5 ( 25 13 + 48 5 + 1015 2 ) × 5 t 2 + 5 13 + 12 5 315 2 × 7 35 2 × ( 29 × 5 t 2 9 ) × 13 5 13 2 ( 5 t 1 + 1 ) ) .
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:
E N T M 1 ( H ) = log ( M 1 ( H ) ) 1 M 1 ( H ) log ( 4 4 × 5 t 2 × 5 5 × ( 5 t 1 + 1 ) × 6 6 × ( 8 · 5 t 2 + 2 ) × 6 6 × 21 · 5 t 2 + 7 4 × 7 7 × 29 · 5 t 2 9 2 × 8 8 × 3 · 5 t 1 7 4 ) = log ( M 1 ( H ) ) 1 M 1 ( H ) log ( 2 1 2 ( 71 × 5 t 1 39 ) × 3 1 2 ( 159 × 5 t 2 + 45 ) × 5 5 t + 5 × 7 7 2 ( 29 × 5 t 2 9 ) ) .
E N T M 2 ( H ) = log ( M 2 ( H ) ) 1 M 2 ( H ) log ( 4 4 × 5 t 2 × 6 6 × ( 5 t 1 + 1 ) × 8 8 × ( 8 · 5 t 2 + 2 ) × 9 9 × 21 · 5 t 2 + 7 4 × 12 12 × 29 · 5 t 2 9 2 × 16 16 × 3 · 5 t 1 7 4 ) = log ( M 2 ( H ) ) 1 M 2 ( H ) log 2 818 × 5 t 2 166 × 3 1 2 ( 597 × 5 t 2 33 ) .
E N T F ( H ) = log ( F ( H ) ) 1 F ( H ) log ( 8 8 × 5 t 2 × 13 13 × ( 5 t 1 + 1 ) × 20 20 × ( 8 · 5 t 2 + 2 ) × 18 18 × 21 · 5 t 2 + 7 4 × 25 25 × 29 · 5 t 2 9 2 × 32 32 × 3 · 5 t 1 7 4 ) = log ( F ( H ) ) 1 F ( H ) log ( 2 1 2 ( 2077 × 5 t 2 337 ) × 3 189 × 5 t 2 + 63 × 5 885 × 5 t 2 185 × 13 13 × 5 t 1 + 13 ) .
E N T A Z I ( H ) = log ( A Z I ( H ) ) 1 A Z I ( H ) log ( 8 8 × 5 t 2 × 8 8 × ( 5 t 1 + 1 ) × 8 8 × ( 8 · 5 t 2 + 2 ) × 729 64 729 64 × 21 · 5 t 2 + 7 4 × 1728 125 1728 125 × 29 · 5 t 2 9 2 × 512 27 512 27 × 3 · 5 t 1 7 4 ) = log ( A Z I ( H ) ) 1 A Z I ( H ) log ( 2 1 48,000 ( 87,354,399 × 5 t 2 34,536,779 ) × 3 1 144,000 ( 107,541,411 × 5 t 2 + 4,684,769 ) × 5 2592 125 ( 29 × 5 t 2 9 ) ) .
E N T S O ( H ) = log ( S O ( H ) ) 1 S O ( H ) log ( ( 2 2 ) 2 2 × 5 t 2 × 13 13 × ( 5 t 1 + 1 ) × ( 2 5 ) 2 5 × ( 8 · 5 t 2 + 2 ) × ( 3 2 ) 3 2 × 21 · 5 t 2 + 7 4 × 5 5 × 29 · 5 t 2 9 2 × ( 4 2 ) 4 2 × 3 · 5 t 1 7 4 ) = log ( S O ( H ) ) 1 S O ( H ) log ( 2 ( 387 8 2 + 16 5 ) × 5 t 2 + 4 5 119 8 2 × 3 21 4 2 ( 3 × 5 t 2 + 1 ) × 5 ( 8 5 + 145 2 ) × 5 t 2 + 2 5 45 2 × 13 13 2 ( 5 t 1 + 1 ) ) .
E N T E S O ( H ) = log ( E S O ( H ) ) 1 E S O ( H ) log ( ( 8 2 ) 8 2 × 5 t 2 × ( 5 13 ) 5 13 × ( 5 t 1 + 1 ) × ( 12 5 ) 12 5 × ( 8 · 5 t 2 + 2 ) × ( 18 2 ) 18 2 × 21 · 5 t 2 + 7 4 × 35 35 × 29 · 5 t 2 9 2 × ( 32 2 ) 32 2 × 3 · 5 t 1 7 4 ) = log ( E S O ( H ) ) 1 E S O ( H ) log ( 2 ( 3319 4 2 + 192 5 ) × 5 t 2 + 48 5 1043 4 2 × 3 ( 189 2 + 96 5 ) × 5 t 2 + 24 5 + 63 2 × 5 ( 25 13 + 48 5 + 1015 2 ) × 5 t 2 + 5 13 + 12 5 315 2 × 7 35 2 × ( 29 × 5 t 2 9 ) × 13 5 13 2 ( 5 t 1 + 1 ) ) .
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.
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 ) ,
E N T S O ( S ( M , t ) ) > E N T M 1 ( S ( M , t ) ) > E N T F ( S ( M , t ) ) > E N T E S O ( S ( M , t ) ) > E N T M 2 ( S ( M , t ) ) > E N T A Z I ( S ( M , t ) ) ;
for graph S ( N , t ) ,
E N T S O ( S ( N , t ) ) > E N T M 1 ( S ( N , t ) ) > E N T F ( S ( N , t ) ) > E N T E S O ( S ( N , t ) ) > E N T A Z I ( S ( N , t ) ) > E N T M 2 ( S ( N , t ) ) .

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.

Author Contributions

Conceptualization, S.-A.X., J.-D.S. and J.-B.L.; data curation, S.-A.X. and J.-D.S.; formal analysis, S.-A.X., J.-D.S. and J.-B.L.; funding acquisition, S.-A.X. and J.-B.L.; investigation, S.-A.X., J.-D.S. and J.-B.L.; methodology, S.-A.X., J.-D.S. and J.-B.L.; project administration, J.-B.L.; resources, S.-A.X. and J.-B.L.; software, S.-A.X. and J.-D.S.; supervision, J.-B.L.; validation, S.-A.X. and J.-B.L.; visualization, S.-A.X. and J.-D.S.; writing—original draft, S.-A.X., J.-D.S. and J.-B.L.; writing—review and editing, S.-A.X. and J.-B.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external.

Data Availability Statement

All data generated or analyzed during this study are included in this published article.

Conflicts of Interest

The authors declare no conflicts of interest.

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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 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.
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Figure 2. The generalized Sierpiński graphs S ( M , 1 ) , S ( M , 2 ) and S ( M , 3 ) , where M is depicted in Figure 1.
Figure 2. The generalized Sierpiński graphs S ( M , 1 ) , S ( M , 2 ) and S ( M , 3 ) , where M is depicted in Figure 1.
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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.
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.
Fractalfract 09 00190 g003
Figure 4. Graphical representation of entropy values of S ( M , t ) and S ( N , t ) for six degree-based topological indices.
Figure 4. Graphical representation of entropy values of S ( M , t ) and S ( N , t ) for six degree-based topological indices.
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Table 1. Edge distribution of G.
Table 1. Edge distribution of G.
( d u , d v ) (2,4)(3,3)(3,4)(4,4)
f G ( d u , d v ) 4 t + 8 3 4 4 t 6 4 t 7 3
Table 2. Edge distribution of H.
Table 2. Edge distribution of H.
( d u , d 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 21 · 5 t 2 + 7 4 29 · 5 t 2 9 2 3 · 5 t 1 7 4
Table 3. Entropy values of E N T f for the generalized Sierpiński graphs S ( M , t ) .
Table 3. Entropy values of E N T f for the generalized Sierpiński graphs S ( M , t ) .
f t = 2 t = 3 t = 4 t = 5 t = 6 t = 7 t = 8 t = 9 t = 10
M 1 3.21364.64946.04797.43728.824210.210711.597112.983414.3697
M 2 3.194.62886.02887.41858.805710.192211.578512.964814.3511
F3.20244.64086.04047.438.817110.203611.5912.976314.3626
A Z I 3.17774.61756.01777.40758.794710.181211.567612.953914.3402
S O 3.21484.65076.04927.43858.825610.21211.598412.984714.371
E S O 3.20014.63846.0387.42768.814810.201311.587612.973914.3602
Table 4. Entropy values of E N T f for the generalized Sierpiński graphs S ( N , t ) .
Table 4. Entropy values of E N T f for the generalized Sierpiński graphs S ( N , t ) .
f t = 2 t = 3 t = 4 t = 5 t = 6 t = 7 t = 8 t = 9 t = 10
M 1 3.57355.21496.83088.441510.051211.660713.270114.879516.489
M 2 3.54035.17986.79578.406410.016111.625613.235114.844516.454
F3.54825.1896.8058.415710.025411.634913.244314.853816.4632
A Z I 3.54325.18296.79888.409610.019311.628713.238214.847616.4571
S O 3.5745.21576.83168.442310.05211.661513.270914.880416.4898
E S O 3.5475.18746.80338.41410.023711.633213.242614.852116.4615
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Xu, S.-A.; Si, J.-D.; Liu, J.-B. Entropiesand Degree-Based Topological Indices of Generalized Sierpiński Graphs. Fractal Fract. 2025, 9, 190. https://doi.org/10.3390/fractalfract9030190

AMA Style

Xu S-A, Si J-D, Liu J-B. Entropiesand Degree-Based Topological Indices of Generalized Sierpiński Graphs. Fractal and Fractional. 2025; 9(3):190. https://doi.org/10.3390/fractalfract9030190

Chicago/Turabian Style

Xu, Si-Ao, Jia-Dong Si, and Jia-Bao Liu. 2025. "Entropiesand Degree-Based Topological Indices of Generalized Sierpiński Graphs" Fractal and Fractional 9, no. 3: 190. https://doi.org/10.3390/fractalfract9030190

APA Style

Xu, S.-A., Si, J.-D., & Liu, J.-B. (2025). Entropiesand Degree-Based Topological Indices of Generalized Sierpiński Graphs. Fractal and Fractional, 9(3), 190. https://doi.org/10.3390/fractalfract9030190

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