Communications on Applied Nonlinear Analysis 

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The M-polynomial of Schreier Graphs of the Basilica and Grigorchuk 

Groups: Comparative Evaluation 

  

Mr.V. Rajkumar1, Dr. B. Sivakumar2, Dr. Nur Idayu Alimon3 

1 Department of Mathematics, Rajalakshmi Engineering College, Chennai – 602 105, rajkumar.v@rajalakshmi.edu.in 

2 Department of Mathematics, Sri Sivasubramaniya Nadar College of Engineering, Chennai -603 110. 

sivakumarb@ssn.edu.in 

3     College of Computing, Informatics and Mathematics, Universiti Teknologi MARA, Johor Branch, Pasir Gudang 

Campus,81750, Johor, Malaysia; idayualimon@uitm.edu.my 

 

Article History: 

Received: 10-04-2024 

Revised: 26-05-2024 

Accepted: 15-06-2024 

Abstract:  

A focus for comprehending the complex structures present in self-similar groups has been 

the study of Schreier graphs related to the Basilica and Grigorchuk group in recent years. 

Basic to group theory, Schreier graphs give a geometric picture of these groups’ actions 

on sets and shed light on the connectivity and symmetry characteristics of these groups. 

This paper examines the M-polynomial of Schreier graphs of the Basilica and Grigorchuk 

groups, examining its consequences for topological indices and comparing the 

determined values.  

Keywords: M-polynomial, Topological indices ,Schreier Graph, Basilica Group 

Grigorchuk Group 

 

1. Introduction  

  Schreier graphs emerge as a fundamental representation of the actions performed by finitely 

generated groups on sets, offering a graphical depiction of these group actions. Particularly within 

the monarchy of automaton groups, Schreier graphs stand out as a natural focus of investigation. 

Finite invertible automata, functioning as finite input/output devices, intricately capture the 

sequential operations executed by a set of states on a finite alphabet. These automata, in turn, 

correspond to automorphisms of rooted regular trees, inducing bijections that collectively form what 

is known as an automaton group (for more details about automaton group please see [1-2]. 

The dynamic behaviour of an automaton group when acting upon a rooted tree manifests in 

the preservation of the hierarchical structure inherent in the tree's levels. By restricting this action to 

a specific level of the tree, one can construct a unique Schreier graph corresponding to that level. As 

a result, each automata group generates an unlimited number of finite Schreier graphs, each of which 

replicates the group's actions on the finite levels of the corresponding rooted regular tree. Within the 

empire of automaton groups laid captivating instances characterized by their unconventional and 

sometimes enigmatic properties. Notably, among these is the Basilica group, a pioneering example 

showcasing intriguing traits. The Basilica group is a unique entity that is produced by a three-state 

automata and is derived from the work of R. Grigorchuk and A. Zuk [3]. This group serves as a 

noteworthy illustration within the broader spectrum of automaton groups, highlighting the richness 

and diversity inherent in their structures. 

mailto:rajkumar.v@rajalakshmi.edu.in


Communications on Applied Nonlinear Analysis 

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A finitely produced torsion group and the first known example of a finitely generated group 

of intermediate growth are obtained from the Grigorchuk group, which also provides the simplest 

solution to the Burnside question. For more details and additional references, see [4] and [5]. 

The M- polynomial is defined by S. Klavzar or E. Deutsch in 2015. One important component in the 

monarchy of degree-oriented topological indices is the M-polynomial. It is the most widely used 

general progressive topological polynomial. Additionally, it has many potential applications is 

various field, which they can be found in [6-10]. 

The M-polynomial of the Schreier graphs associated with the actions of the two well-known 

automorphism groups of the binary rooted tree, Basilica group and Grigorchuk group is investigated 

in this paper. These groups can be associated with a compact limit space that is homeomorphic to the 

Basilica fractal since it can be described as an iterated monodromy group of the complex polynomial 

z2 – 1.  This is the first example of an amenable group that does not belong to a subexponentially 

pliable group [11]. This group has been demonstrated to have strong connections with complex 

dynamics and prophinite group theory [12]. The self-similarity value of some limit objects related to 

these groups reflects a fractal nature [8]. 

The primary conclusion of section 2 is the M-polynomial of the Basilica group's and 

Grigorchuk group's Schreier graphs, from which various degree-based topological indices are 

constructed. In Section 3, the selected topological are calculated and we make a comparison between 

the topological indices obtained from the M-polynomial and the estimated values of a few selected 

topological indices. Finally, the conclusion of this work is done in section 4.  

2. Main Results  

In this section the M-polynomial of Schreier graphs of the Basilica group is derived. Using 

that some degree based topological indices is derived.  

2.1 The M-polynomial of Schreier graphs (without loops) of the Basilica group 

The Basilica group is an automorphism collection that is self-similar and generated by the 

elements 𝑎 = 𝑒(𝑏, 𝑖𝑑)  and 𝑏 =∈ (𝑎, 𝑖𝑑) of the rooted binary tree. You may find the substitution 

rules in [13], which are used to generate the corresponding Schreier graphs iteratively. For n>1, let 

𝐵𝑛 be the notation for the Schreier graphs of the Basilica group, which are thought to be loop-free 

because the calculations for graphs with loops are trivial. Figure 1 display these graphs, as shown in 

[14]. 

Definition 2.1 Let 𝐵𝑛 = 𝐵𝑛 (ɣ, Ȅ) be a connected graph where ɣ is the set of vertices and Ȅ is the set 

of edges of 𝐵𝑛. The M – Polynomial of graph 𝐵𝑛 is defined as 𝑓(𝑤, 𝑧) = 𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z) =

∑ |Ȅ𝑖𝑗(𝐵𝑛)|𝑤𝑖𝑧𝑗
𝕀≤𝑖≤𝑗≤𝕁 , where 𝕀 = 𝑀𝑖𝑛{𝑑(𝓋) 𝓋𝜖ɣ(𝐵𝑛)⁄ }; 𝕁 = 𝑀𝑎𝑥{𝑑(𝓋) 𝓋𝜖ɣ(𝐵𝑛)⁄ }, and Ȅ𝑖𝑗(𝐵𝑛) is 

the edge 𝓋𝓊 𝜖 Ȅ for which {𝑑(𝓋), 𝑑(𝓊)} = {𝑖, 𝑗}. Where 𝑑(𝓋) and 𝑑(𝓊) are degree of a vertices 𝓋 

and 𝓊 respectively.  

On the bases of growth of the Schreier graphs (without loops) of Basilica groups having 

edges partition Ȅ𝑖𝑗 = {𝓋𝓊 𝜖 Ȅ(Ɠ) 𝑑(𝓋) = 𝑖, 𝑑(𝓊) = 𝑗⁄ } is the set of all edges with end degrees 

𝑑(𝓋) = 𝑖, 𝑑(𝓊) = 𝑗). On the bases of degrees, we divide the edges into two partitions for 𝐵𝑛, 𝑛 > 1.  



Communications on Applied Nonlinear Analysis 

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Ȅ24 = {𝓋𝓊 𝜖 Ȅ(Ɠ) 𝑑(𝓋) = 2, 𝑑(𝓊) = 4⁄ } 

Ȅ44 = {𝓋𝓊 𝜖 Ȅ(Ɠ) 𝑑(𝓋) = 4, 𝑑(𝓊) = 4⁄ }. 

 

 

 

Fig. 1 Structure of  𝐵𝑛 for some 𝑛 



Communications on Applied Nonlinear Analysis 

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The following Table 1 depicts the selected topological indices for this study and its derivation 

formulas [6] through M-polynomial.   

Table 1. Degree based M – Polynomial 

Topological index Degree based Derivation 

First Zagreb index (𝑀1) ∑ 𝑑(𝓋) + 𝑑(𝓊)

𝑒=𝓋𝓊∈Ȅ

 (𝐷𝑤 + 𝐷𝑧) (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z))|
𝑤=𝑧=1

 

Second Zagreb index (𝑀2) ∑ 𝑑(𝓋)𝑑(𝓊)

𝑒=𝓋𝓊∈Ȅ

 (𝐷𝑤𝐷𝑧) (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛 , w, z))|
𝑤=𝑧=1

 

Modified second Zagreb 

index ( 𝑀2
𝑚) 

∑
1

𝑑(𝓋)𝑑(𝓊)
𝑒=𝓋𝓊∈Ȅ

 (𝑆𝑤𝑆𝑧) (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z))|
𝑤=𝑧=1

 

Hyper Zagreb index (HM) ∑ [𝑑(𝓋) + 𝑑(𝓊)]2

𝑒=𝓋𝓊∈Ȅ

 (𝐷𝑤 + 𝐷𝑧)2 (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛 , w, z))|
𝑤=𝑧=1

 

Forgotten index (F) ∑ 𝑑(𝓋)2 + 𝑑(𝓊)2

𝑒=𝓋𝓊∈Ȅ

 (𝐷𝑤
2 + 𝐷𝑧

2) (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛 , w, z))|
𝑤=𝑧=1

 

Inverse sum index (I) ∑
𝑑(𝓋)𝑑(𝓊)

𝑑(𝓋) + 𝑑(𝓊)
𝑒=𝓋𝓊∈Ȅ

 (𝑆𝑤𝐽 𝐷𝑤𝐷𝑧) (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛 , w, z))|
𝑤=1

 

Augmented Zagreb index 

(A) 
∑ (

𝑑(𝓋)𝑑(𝓊)

𝑑(𝓋) + 𝑑(𝓊) − 2
)

3

𝑒=𝓋𝓊∈Ȅ

 (𝑆𝑤𝑄−2𝐽 𝐷𝑤
3𝐷𝑧

3) (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z))|
𝑤=1=11

 

Note that 𝐷𝑤 = 𝑤
𝜕

𝜕𝑤
𝑓(𝑤, 𝑧) , 𝐷𝑧 = 𝑧

𝜕

𝜕𝑧
𝑓(𝑤, 𝑧), 𝑆𝑤 = ∫

𝑓(𝑡,𝑧)

𝑡
𝑑𝑡

𝑤

0
, 𝑆𝑧 = ∫

𝑓(𝑤,𝑡)

𝑡
𝑑𝑡

𝑧

0
,  

𝐽(𝑓(𝑤, 𝑧)) = 𝑓(𝑤, 𝑤),𝑄∝(𝑓(𝑤, 𝑧)) = 𝑤∝𝑓(𝑤, 𝑧). 

Theorem 2.1. Let 𝐵𝑛 be Schreier graph (without loops) of Basilica group. Then the M – Polynomial 

of 𝐵𝑛 is 𝑓(𝑤, 𝑧) = 𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z) = (2𝑛)𝑤2𝑧4 + (2𝑛−1)𝑤4𝑧4. 

Proof. From the following Table 2 we can understand the edge growth pattern and edge partition of 

Schreier graph of Basilica group. 

Table 2. Edge Growth pattern and edge partition of Schreier graph of Basilica group 

Basilica 

Group (𝑩𝒏) 

Number of 

vertices 

Number of 

edges 
Edge growth pattern 

Edge Partitions 

(2,4) (4,4) 

B2 4 6 (22)  +  (21) 4 2 

B3 8 12 (23)  +  (22) 8 4 

B4 16 24 (24)  +  (23) 16 8 

B5 32 48 (25)  +  (24) 32 16 

B6 64 96 (26)  +  (25) 64 32 

. . . . . . 

. . . . . . 

Bn 2𝑛   (2𝑛)  +  (2𝑛−1) 2𝑛 2𝑛−1 

From Table 1, it is clear that the number of edges in the above two categories are |Ȅ24| = 2𝑛   and  

|Ȅ44| = 2𝑛−1.   

Now, by the definition of M-polynomial, we have  



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𝑓(𝑤, 𝑧) = 𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z) = ∑ |Ȅ𝑖𝑗(𝐵𝑛)|𝑤𝑖𝑧𝑗

𝕀≤𝑖≤𝑗≤𝕁
 

                                                                   = |Ȅ24|𝑤2𝑧4 + |Ȅ44|𝑤4𝑧4 

                                                    ∴ 𝑓(𝑤, 𝑧) = (2𝑛)𝑤2𝑧4 + (2𝑛−1)𝑤4𝑧4. □ 

 

Fig. 2 Surface plot of M-polynomial of 𝐵𝑛. 

Theorem 2.2 Let 𝐵𝑛 be the Schreier graph of Basilica group. Then the M-polynomials for the 

topological indices of 𝐵𝑛 for 𝑛 > 1 are  

a) 𝑀1(𝐵𝑛) = 5 × 2𝑛+1 

b) 𝑀2(𝐵𝑛) = 2𝑛+4 

c) 𝑀2
𝑚(𝐵𝑛) = 5 × 2𝑛−5 

d) 𝐴𝑍(𝐵𝑛) = 3 × 2𝑛+1 

e) 𝐻𝑀(𝐵𝑛) = 17 × 2𝑛+2 

f) 𝐹(𝐵𝑛) = 3 × 2𝑛+3 

g)  𝐼(𝐵𝑛) =
7

3
2𝑛. 

Proof. For (a), 𝑀1(𝐵𝑛) = (𝐷𝑤 + 𝐷𝑧) (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z))|
𝑤=𝑧=1

 

= |(2𝑛). 2𝑤2𝑧4 + (2𝑛). 4𝑤2𝑧4 + (2𝑛−1)4𝑤4𝑧4 + (2𝑛−1)4𝑤4𝑧4|𝑤=𝑧=1 = 5 × 2𝑛+1. 

For (b), 𝑀2(𝐵𝑛) = (𝐷𝑤𝐷𝑧) (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z))|
𝑤=𝑧=1

 

First 

 𝐷𝑍 (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = (2𝑛)4𝑤2𝑧4 + (2𝑛−1)4𝑤4𝑧4 

Now, 𝐷𝑤𝐷𝑧 (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = 𝐷𝑤[(2𝑛) × 4 × 𝑤2𝑧4 + (2𝑛−1) × 4 × 𝑤4𝑧4] 

= (2𝑛) × 4 × 2 × 𝑤2𝑧4 + (2𝑛−1) × 4 × 4 × 𝑤4𝑧4 = 2𝑛+4. 



Communications on Applied Nonlinear Analysis 

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For (c), 𝑀2
𝑚(𝐵𝑛) = (𝑆𝑤𝑆𝑧) (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z))|

𝑤=𝑧=1
,   𝑆𝑤 = ∫

𝑓(𝑡,𝑧)

𝑡
𝑑𝑡

𝑤

0
= ∫

(2𝑛
)𝑡2𝑧4+(2𝑛−1)𝑡4𝑧4

𝑡
𝑑𝑡

𝑤

0
  

∴ 𝑆𝑤 = (2𝑛
)𝑧4 𝑤2

2
+ (2𝑛−1)𝑧4 𝑤4

4
,  𝑆𝑤𝑆𝑧 = ∫

𝑓(𝑤,𝑡)

𝑡
𝑑𝑡

𝑧

0
= ∫

[(2𝑛
)𝑡4𝑤2

2
+(2𝑛−1)𝑡4𝑤4

4
]

𝑡

𝑧

0
𝑑𝑡 = (2𝑛

)
𝑧4

4

𝑤2

2
+ (2𝑛−1

)
𝑧4

4

𝑤4

4
 

∴ 𝑀2
𝑚(𝐵𝑛) = (𝑆𝑤𝑆𝑧) (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z))|

𝑤=𝑧=1
=

2𝑛

8
+

2𝑛−1

16
= 2𝑛−3 + 2𝑛−5 = 5 × 2𝑛−5 

For (d), to find (𝑆𝑤
3𝑄−2𝐽 𝐷𝑤

3𝐷𝑧
3) (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z))|

𝑤=1
 

𝐷𝑍 (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = (2𝑛)4𝑤2𝑧4 + (2𝑛−1)4𝑤4𝑧4, 𝐷𝑍
2 (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = [(2𝑛

) × 16 × 𝑤2𝑧4] +

[(2𝑛−1
) × 16 × 𝑤4𝑧4], 𝐷𝑍

3 (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = [(2𝑛
) × 64 × 𝑤2𝑧4] + [(2𝑛−1

) × 64 × 𝑤4𝑧4] 

𝐷𝑤𝐷𝑍
3 (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = [(2𝑛

) × 128 × 𝑤2𝑧4] + [(2𝑛−1
) × 256 × 𝑤4𝑧4] 

𝐷𝑤
2𝐷𝑍

3
(𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = [(2𝑛

) × 256 × 𝑤2𝑧4] + [(2𝑛−1
) × 1024 × 𝑤4𝑧4] 

𝐷𝑤
3𝐷𝑍

3
(𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = [(2𝑛

) × 512 × 𝑤2𝑧4] + [(2𝑛−1
) × 4096 × 𝑤4𝑧4] 

  𝐽𝐷𝑤
3𝐷𝑍

3
(𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z))    = [(2𝑛

) × 512 × 𝑤6] + [(2𝑛−1
) × 4096 × 𝑤8] 

𝑄−2𝐽𝐷𝑤
3𝐷𝑍

3
(𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = 𝑤−2 {[(2𝑛

) × 512 × 𝑤6] + [(2𝑛−1
) × 4096 × 𝑤8]} 

𝑆𝑤𝑄−2𝐽𝐷𝑤
3𝐷𝑍

3
(𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = ∫ (

[(2𝑛
)×512×𝑡6]+[(2𝑛−1

)×4096×𝑡8]

𝑡
) 𝑑𝑡

𝑤

0
  = (2𝑛

) × 512 × 𝑤6

6
+ (2𝑛−1

) ×

4096 × 𝑤8

8
.  

𝑆𝑤
2𝑄−2𝐽𝐷𝑤

3
𝐷𝑍

3

(𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = ∫
(2𝑛

)×512×𝑡6

6
+(2𝑛−1

)×4096×𝑡8

8

𝑡

𝑤

0
𝑑𝑡  = (2𝑛

) × 512 × 𝑤6

36
+ (2𝑛−1

) × 4096 ×
𝑤8

64
. 

𝑆𝑤
3𝑄−2𝐽𝐷𝑤

3
𝐷𝑍

3

(𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = ∫
(2𝑛

)×512×𝑡6

36
+(2𝑛−1

)×4096×𝑡8

64

𝑡

𝑤

0
𝑑𝑡 = (2𝑛

) × 512 × 𝑤6

216
+ (2𝑛−1

) ×

4096 × 𝑤8

512
.    

 

∴ 𝐴𝑍(𝐵𝑛) = (𝑆𝑤
3𝑄−2𝐽 𝐷𝑤

3𝐷𝑧
3) (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z))|

𝑤=1
= [(2𝑛

) × 2] + [(2𝑛−1
) × 8] = 3 × 2𝑛+1

 

For (e), to find  𝐻𝑀(𝐵𝑛) = (𝐷𝑤 + 𝐷𝑧)2 (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z))|
𝑤=𝑧=1

, (𝐷𝑤 + 𝐷𝑧)2 = (𝐷𝑤 + 𝐷𝑧)(𝐷𝑤 +

𝐷𝑧)𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z) = (𝐷𝑤 + 𝐷𝑧) {(2𝑛
). 2𝑤2𝑧4 + (2𝑛

). 4𝑤2𝑧4 + (2𝑛−1)4𝑤4𝑧4 + (2𝑛−1)4𝑤4𝑧4} 

(𝐷𝑤 + 𝐷𝑧)2 = |9𝑤2𝑧42𝑛+2 + 𝑤4𝑧42𝑛+5
|

𝑤=1
,  ∴  𝐻𝑀(𝐵𝑛) = 17 × 2𝑛+2. 

For (f)  

To find  𝐹(Ɠ) = (𝐷𝑤
2 + 𝐷𝑧

2) (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z))|
𝑤=𝑧=1

 



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𝐷𝑤 (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = 𝐷𝑤[(2𝑛)𝑤2𝑧4 + (2𝑛−1)𝑤4𝑧4] = [2𝑛+1 × 𝑤2𝑧4] + [2𝑛+1 × 𝑤4𝑧4] 

𝐷𝑤
2 (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = [2𝑛+2 × 𝑤2𝑧4] + [2𝑛+3 × 𝑤4𝑧4], 𝐷𝑍 (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = [2𝑛+2 ×

𝑤2𝑧4] + [2𝑛+1 × 𝑤4𝑧4] 

𝐷𝑍
2 (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = [2𝑛+4 × 𝑤2𝑧4] + [2𝑛+3 × 𝑤4𝑧4] 

∴ (𝐷𝑤
2 + 𝐷𝑧

2) (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = 3 × 2𝑛+3 

∴ 𝐹(𝐵𝑛) = 3 × 2𝑛+3. 

For (g)  

To find  𝐼(𝐵𝑛) = (𝑆𝑤𝐽 𝐷𝑤𝐷𝑧) (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z))|
𝑤=1

 

From (ii) 𝐷𝑤𝐷𝑧 (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = (2𝑛) × 4 × 2 × 𝑤2𝑧4 + (2𝑛−1) × 4 × 4 × 𝑤4𝑧4 

𝐽𝐷𝑤𝐷𝑧 (𝑀𝑝𝑜𝑙𝑦(𝐵𝑛, w, z)) = [8 × 2𝑛𝑤6] + [16 × 2𝑛−1𝑤8], (𝑆𝑤𝐽 𝐷𝑤𝐷𝑧) (𝑀𝑝𝑜𝑙𝑦(Ɠ, w, z)) =

∫ [
[8×2𝑛𝑡6]+[16×2𝑛−1𝑡8]

𝑡
] 𝑑𝑡

𝑤

0
=

8×2𝑛𝑤6

6
+

16×2𝑛−1𝑤8

8
 

∴ 𝐼(Ɠ) = (𝑆𝑤𝐽 𝐷𝑤𝐷𝑧) (𝑀𝑝𝑜𝑙𝑦(Ɠ, w, z))|
𝑤=1

=
7

3
2𝑛. □ 

 

2.2 The M-polynomial of Schreier graphs (with loops) of the Basilica group 

 In this section the Schreier graph of the Basilica group with loops considered and the M-

polynomial for the graph is derived in the following Theorem 2.3. The surface plot of this M-

polynomial can be seen in Figure 4. Moreover, by the use of this M-polynomial the chosen 

topological indices are derived in Theorem 2.4.   

 Theorem 2.3. Let 𝐵𝑛
∗ be Schreier graph (with loops) of Basilica group. Then the M – Polynomial of 

𝐵𝑛
∗ is 𝑓(𝑤, 𝑧) = 𝑀𝑝𝑜𝑙𝑦(𝐵𝑛

∗, w, z) = (2𝑛+1)𝑤4𝑧4. 

Proof. From Figure 1, degree of each vertex in 𝐵𝑛
∗ is four if we consider the loops. Hence the 

Schreier graph of Basilica group is a 4- regular graph. So, the number vertices in the graph are 2𝑛 

and the number of edges is 2𝑛+1. 

That is, |𝑉(𝐵𝑛⬚
∗ )| = 2𝑛, |𝐸(𝐵𝑛

∗)| = 2𝑛+1.  

Since 𝐵𝑛
∗ is a 4-regular graph, there will be only one edge partition which is (4, 4) and therefore 

|Ȅ44| = 2𝑛+1 

Therefore, we have by the definition 2.1, 

𝑓(𝐵𝑛
∗, 𝑤, 𝑧) = 𝑀𝑝𝑜𝑙𝑦(𝐵𝑛

∗, w, z) = ∑ |Ȅ𝑖𝑗(𝐵𝑛
∗)|𝑤𝑖𝑧𝑗

𝕀≤𝑖≤𝑗≤𝕁  = |Ȅ44|𝑤4𝑧4 



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                                                    ∴ 𝑓(𝐵𝑛
∗, 𝑤, 𝑧) = (2𝑛+1)𝑤4𝑧4. 

 

Fig. 3 Surface plot of M-polynomial of 𝐵𝑛
∗  

Theorem 2.4 Let 𝐵𝑛
∗ be the Schreier graph (including loops) of Basilica group. Then the topological 

indices of 𝐵𝑛
∗ ,  𝑛 > 1 using the M-polynomial are  

a) 𝑀1(𝐵𝑛
∗) = 2𝑛+4 

b) 𝑀2(𝐵𝑛
∗) = 2𝑛+5 

c) 𝑀2
𝑚(𝐵𝑛

∗) = 2𝑛−3 

d) 𝐴𝑍(𝐵𝑛
∗) = 2𝑛+4 

e) 𝐻𝑀(𝐵𝑛
∗) = 2𝑛+7 

f) 𝐹(𝐵𝑛
∗) = 2𝑛+6 

g) 𝐼(𝐵𝑛
∗) = 2𝑛+2. 

Proof. These results can be proved using Theorem 2.2. □ 

2.3 The M-polynomial of Schreier graphs (without loops) of the Grigorchuk group 

Let Γ𝑛 be the Schreier graph of the Grigorchuk group. In finding the M-polynomial, this 

graph is considered as unlabelled and without loops. The graph Γ𝑛 for 𝑛 > 1 can be seen in the 

following Figure 4.  

 

Figure 4. Structure of  Γ𝑛 for 𝑛 = 1,2,3. 

Theorem 2.5. Let Γ𝑛 be Schreier graph (without loops) of Grigorchuk group. Then the M – 

polynomial of Γ𝑛 is 𝑓(Γ𝑛, 𝑤, 𝑧) = 𝑀𝑝𝑜𝑙𝑦(Γ𝑛, w, z) = 2𝑤1𝑧3 + (2𝑛 + 2𝑛−1 − 4)𝑤3𝑧3. 



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Proof. From the following Table 3, one can understand the edge growth pattern and edge partition of 

Schreier graph of Basilica group. 

Table 3. Edge Growth pattern and edge partition of Schreier graph of Grigorchuk group 

Grigorchuk 

group 

No. of 

vertex (x) 
No of Edges(y) Edge growth pattern 

Edge Partitions 

(1,3) (3,3) 

Γ2 4 4 (21) + (21) 2 2 

Γ3 8 10 (21)+ (2)3 2 8 

Γ4 16 22 (21) + (24 + 4) 2 20 

Γ5 32 46 (21) + (25 + 12) 2 44 

Γ6 64 94 (21) + (26 + 28) 2 92 

. . . . . . 

. . . . . . 

Γ𝑛 2𝑛 3. 2𝑛−1 − 2  (21) + 2𝑛 + (2𝑛−1 − 4) 21 2𝑛 + 2𝑛−1 − 4 

From Table 3, it is clear that the number of edges in the above two category are |Ȅ13| = 2   

and|Ȅ33| = 2𝑛 + 2𝑛−1 − 4.   

Now, by the definition 2.1, we have  

𝑓(Γ𝑛, 𝑤, 𝑧) = 𝑀𝑝𝑜𝑙𝑦(Γ𝑛, w, z) = ∑ |Ȅ𝑖𝑗(Γ𝑛)|𝑤𝑖𝑧𝑗

𝕀≤𝑖≤𝑗≤𝕁
 

                                                                   = |Ȅ13|𝑤1𝑧3 + |Ȅ33|𝑤3𝑧3 

                                                    ∴ 𝑓(Γ𝑛, 𝑤, 𝑧) = 2𝑤1𝑧3 + (2𝑛 + 2𝑛−1 − 4)𝑤3𝑧3. □ 

 

Figure 5. Surface plot of M-polynomial of Γ𝑛. 

Theorem 2.6 Let Γ𝑛 be the Schreier graph of Grigorchuk group. Then the M-polynomials for the 

topological indices of Γ𝑛 , 𝑛 > 1 are  

a) 𝑀1(Γ𝑛)= 6 (2𝑛 + 2𝑛−1) − 16. 

b) 𝑀2(Γ𝑛) = 9(2𝑛 + 2𝑛−1) − 30. 



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c) 𝑀2
𝑚(Γ𝑛) =

2𝑛+2𝑛−1+2

9
. 

d) 𝐴𝑍(Γ𝑛) =
729(2𝑛+2𝑛−1)−2484

64
. 

e) 𝐻𝑀(Γ𝑛) = 36(2𝑛 + 2𝑛−1) − 112. 

f) 𝐹(Γ𝑛) = 18(2𝑛 + 2𝑛−1) − 52. 

g)  𝐼(Γ𝑛) =
3(2𝑛+2𝑛−1)−9

2
. 

Proof. It can be verified using Theorem 2.2. □ 

2.4 The M-polynomial of Schreier graphs (with loops) of the Grigorchuk group 

 Let Γ𝑛
∗ be the Schreier graph (including loops) of the Grigorchuk group. In finding the M-

polynomial, this graph is considered as unlabelled and with loops. From Figure 4, it is clear that the 

number of vertices  |𝑉(Γ𝑛
∗ )| = 2𝑛 and |𝐸(Γ𝑛

∗ )| = 5. 2𝑛−1 + 2 when we consider the loops.    

Theorem 2.7. Let Γ𝑛
∗ be Schreier graph (with loops) of Grigorchuk group. Then the M – polynomial 

of Γ𝑛
∗  is 𝑓(Γ𝑛

∗ , 𝑤, 𝑧) = 𝑀𝑝𝑜𝑙𝑦(Γ𝑛
∗ , w, z) = 2𝑤5𝑧7 + 6𝑤7𝑧7 + (5. 2𝑛−1 − 6)𝑤5𝑧5. 

Proof. From the Figure 4, it is clear that there are three different categories of edge partitions of 

Schreier graph  Γ𝑛
∗ of Grigorchuk group and they are 

 |Ȅ57| = 2, |Ȅ77| = 6  and|Ȅ55| = 5. 2𝑛−1 − 6.   

 Now, by the definition of 2.1, we have  

𝑓(Γ𝑛
∗ , 𝑤, 𝑧) = 𝑀𝑝𝑜𝑙𝑦(Γ𝑛

∗ , w, z) = ∑ |Ȅ𝑖𝑗(Γ𝑛
∗ )|𝑤𝑖𝑧𝑗

𝕀≤𝑖≤𝑗≤𝕁
 

                                                                   = |Ȅ57|𝑤5𝑧7 + |Ȅ55|𝑤5𝑧5 + |Ȅ77|𝑤7𝑧7 

                                                    ∴ 𝑓(Γ𝑛
∗ , 𝑤, 𝑧) = 2𝑤5𝑧7 + 6𝑤7𝑧7 + (5. 2𝑛−1 − 6)𝑤5𝑧5. □ 

 

Figure 6. 3-D graph of M-polynomial of Γ𝑛
∗. 



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Theorem 2.8 Let Γ𝑛
∗  be the Schreier graph (including loops) of Grigorchuk group. Then the 

topological indices of Γ𝑛
∗, 𝑛 > 1  are  

a) 𝑀1(Γ𝑛
∗ )= (25 × 2𝑛)  − 62. 

b) 𝑀2(Γ𝑛
∗ ) = (125 × 2𝑛−1) + 214. 

c) 𝑀2
𝑚(Γ𝑛

∗ ) =
2𝑛

10
− 0.0605. 

d) 𝐴𝑍(Γ𝑛
∗ ) = (152.5875 × 2𝑛−1) + 311.148. 

e) 𝐻𝑀(Γ𝑛
∗ ) = (250 × 2𝑛) + 864. 

f) 𝐹(Γ𝑛
∗ ) = (125 × 2𝑛) + 436. 

g) 𝐼(Γ𝑛
∗ ) = (12.5 × 2𝑛−1) + 864. 

Proof. It can be verified using Theorem 2.2. □ 

3. Calculations and Comparative analysis 

 In this section the values of the selected topological indices calculated by using the results in 

the previous sections and comparative study done based on the determined results.  Table 4 presents 

calculated values of various topological indices for 𝐵𝑛. These indices provide various measures of 

the structure and properties of 𝐵𝑛 at different levels.  

For example, 𝑀1(𝐵𝑛) and 𝑀2(𝐵𝑛) provide basic information about the size of the graph and 

its connectivity, while indices like 𝐴𝑍(𝐵𝑛), 𝐻𝑀(𝐵𝑛), and 𝐼(𝐵𝑛) give understandings into more 

specific properties related to vertex independence, distances, and autonomy. 

Table 4. Calculated values of topological indices of 𝐵𝑛. 

 

     

 

 

 

 

 

When we display these numbers on a line graph (see Figure 7), we see that when the level of the 

Basilica group rises, all of the indices exhibit consistent patterns of exponential growth. The graph's 

rising complexity is reflected in the indices 𝑀1(𝐵𝑛),𝑀2(𝐵𝑛) and 𝐹(𝐵𝑛), which grow quickly as the 

number of vertices and edges increases exponentially. Similarly, indices like 𝐴𝑍(𝐵𝑛) and 𝐻𝑀(𝐵𝑛) 

also display significant growth, though at a slightly slower pace, indicating the increasing autonomy 

and harmonic complexity of the graph structure. Additionally, the indices 𝑀2
𝑚(𝐵𝑛) and 𝐼(𝐵𝑛) follow 

the general trend of exponential growth but with smaller numerical values. Overall, the line chart 

illustrates the dynamic growth and complexity of the Basilica group's Schreier graph across different 

Topological  

Index 

Basilica group Bn 

 𝑩𝟐 𝑩𝟑 𝑩𝟒 𝑩𝟓 𝑩𝟔 

𝑀1(𝐵𝑛)  40 80 160 320 640 

 𝑀2(𝐵𝑛)  64 128 256 512 1024 

𝑀2
𝑚(𝐵𝑛)  0.62 1.25 2.5 5 10 

𝐴𝑍(𝐵𝑛)  24 48 96 192 384 

𝐻𝑀(𝐵𝑛)  272 544 1088 2176 4352 

𝐹(𝐵𝑛)  96 192 384 768 1536 

𝐼(𝐵𝑛)  9.33 18.67 37.33 74.67 149.33 



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topological indices, highlighting the convoluted interplay between vertices, edges, distances, and 

autonomy within the graph structure as the group level increases. 

 

Fig. 7 Line chart for topological index value of Bn. 

Table 5. Calculated values of topological indices of Bn* 

Topological Index 
Basilica group Bn* 

 B2* B3* B4* B5* B6* 

𝑀1(𝐵𝑛
∗)  64 128 256 512 1024 

 𝑀2(𝐵𝑛
∗)  128 256 512 1024 2048 

𝑀2
𝑚(𝐵𝑛

∗)  0.5 1 2 4 8 

𝐴𝑍(𝐵𝑛
∗)  64 128 256 512 1024 

𝐻𝑀(𝐵𝑛
∗)  512 1024 2048 4096 8192 

𝐹(𝐵𝑛
∗)  256 512 1024 2048 4096 

𝐼(𝐵𝑛
∗)  16 32 64 128 256 

 

Table 5 presents the computed values of various topological indices for the Schreier graph with loops 

in the Basilica group at different levels labelled as 𝐵1
∗ to 𝐵6

∗. A thorough understanding of the 

structural complexity of the graph at various group levels is offered by these indices. Analysis 

reveals that most indicators show a consistent exponential development trend as the group level 

increases. This can be seen in Figure 8. Interestingly, indices such as 𝑀1(𝐵𝑛
∗),  𝑀2(𝐵𝑛

∗), 𝐴𝑍(𝐵𝑛
∗), 

and 𝐹(𝐵𝑛
∗) show strong increases, reflecting the graph's growing vertex and edge network. In 

addition, the Hyper Zagreb index 𝐻𝑀(𝐵𝑛
∗) shows significant increase, indicating increased 

harmonic complexity in the graph structure. Interestingly, however, the indices 𝐼(𝐵𝑛
∗) and 𝑀2

𝑚(𝐵𝑛
∗) 

exhibit exponential development patterns with much smaller numerical values.  

200

1200

2200

3200

4200

5200

B1 B2 B3 B4 B5 B6

Basilica group Bn

Topological Indices of Bn



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Fig. 8 Line chart for calculated values of topological indices of  𝐵𝑛
∗ 

The following Table 6 provides calculated values of various topological indices for Schreier graph of 

the Grigorchuk group at different levels.  

Table 6. Calculated values of topological indices of Γ𝑛 

Topological Index 
Grigorchuk group 

Γ2 Γ3 Γ4 Γ5 Γ6 

𝑀1(Γ𝑛) 20 56 128 272 560 

 𝑀2(Γ𝑛) 24 78 186 402 834 

𝑀2
𝑚(Γ𝑛) 0.8889 1.6 2.88889 5.55556 10.8889 

𝐴𝑍(Γ𝑛) 29.531 98 234.563 507.938 1054.69 

𝐻𝑀(Γ𝑛) 104 320 752 1616 3344 

𝐹(Γ𝑛) -4 44 140 332 716 

𝐼(Γ𝑛) 4.5 14 31.5 67.5 139.5 

 

A 3- D graph (see Figure 9) representing these patterns would give a clear picture of how each 

topological index varies with the Grigorchuk group level, revealing details about the structural 

characteristics and complexity of the related graph at various group sizes.  

200

700

1200

1700

2200

2700

3200

3700

4200

4700

5200

B1* B2* B3* B4* B5* B6*

Basilica group Bn*

Topological indices of Bn*  



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Fig. 9 3-D plot of topological indices of  Γ𝑛. 

Table 7 shows the determined values of several topological indices at different levels (represented by 

Γ2
∗ to Γ6

∗) for the Schreier graph of the Grigorchuk group, including loops. There are distinct trends 

among the indices: when the group level rises, the indices  

𝑀1(Γ𝑛) and  𝑀2(Γ𝑛) show steady growth, signifying an increase in vertices and edges in the graph. 

Although its numerical values are less, the Modified Second Zagreb index displays a similar 

tendency. Especially, the Hyper and Augmented Zagreb indices show significant increases, 

indicating more intricacy and connection in the network structure. The Inverse Sum index gradually 

increases across several group levels, whereas the forgotten index consistently remains positive.  

Table 7. Calculated values of topological indices of  Γ𝑛
∗  

Topological Index 
Grigorchuk group 

𝚪𝟐
∗ 𝚪𝟑

∗ 𝚪𝟒
∗ 𝚪𝟓

∗ 𝚪𝟔
∗ 

𝑀1(Γ𝑛
∗) 38 138 338 738 1538 

 𝑀2(Γ𝑛
∗) 464 714 1214 2214 4214 

𝑀2
𝑚(Γ𝑛

∗) 0.3395 0.7395 1.5395 3.1395 6.3395 

𝐴𝑍(Γ𝑛
∗) 616.32 921.498 1531.848 2752.548 5193.948 

𝐻𝑀(Γ𝑛
∗) 1864 2864 4864 8864 16864 

𝐹(Γ𝑛
∗) 936 1436 2436 4436 8436 

𝐼(Γ𝑛
∗) 889 914 964 1064 1264 

 

The graphical representation of the calculated values for the topological indices of Γ𝑛
∗is presented in 

the Figure 10. In this plot, each axis represent a different aspect: one axis would represent the group 

level (Γ2
∗ to Γ6

∗), another axis would represent the specific topological index being measured, and 

the third axis would represent the corresponding calculated value for that index. The plot would 

consist of multiple lines or surfaces, each representing a different topological index, with points 

along each line or surface indicating the calculated values at different group levels. 

Grigorchuk group Ⴈ1 Grigorchuk group Ⴈ2 Grigorchuk group Ⴈ3

Grigorchuk group Ⴈ4 Grigorchuk group Ⴈ6



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This 3D visualization would provide a comprehensive depiction of how each topological index 

varies with both the group level and the specific measurement being considered, offering a clear and 

intuitive understanding of the evolving characteristics of Γ𝑛
∗. 

 

Fig. 10 Calculated values of topological indices of  Γ𝑛
∗. 

4. Conclusion  

 The study explores into the characterization of self-similar groups through Schreier 

graphs, particularly focusing on the Basilica and Grigorchuk groups. By analyzing the M-polynomial 

of these graphs, the paper explores its implications on various topological indices, providing 

understandings into the connectivity and symmetry properties of these groups. Similarly, the 

calculated values of these indices for the Grigorchuk group Γn and its variant Γn*, offering a 

comparative analysis between the two groups. 

From the calculated values, it is evident that there are discernible patterns in the values of the 

topological indices across different iterations (n) of both groups. For instance, the First Zagreb index 

generally exhibits exponential growth with increasing n for both Basilica and Grigorchuk groups, 

reflecting their complex structural characteristics. 

Moreover, comparing the indices of the original groups with their variants (Bn vs. Bn* and Γn vs. 

Γn*), it is seeming that certain transformations or modifications lead to significant alterations in the 

topological properties, as evidenced by changes in index values. This analysis provides valuable 

perceptions into the topological characteristics of the Basilica and Grigorchuk groups, illuminating 

on their complex structures and behaviors. Such investigations contribute to a deeper understanding 

of self-similar groups and their geometric representations, with potential implications across various 

domains, including group theory, topology, and computational mathematics. 

 

 

Grigorchuk group Ⴈ2
Grigorchuk group Ⴈ3
Grigorchuk group Ⴈ4
Grigorchuk group Ⴈ5
Grigorchuk group Ⴈ6

0
2000
4000
6000
8000

10000
12000
14000
16000
18000



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References  

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