Study on (r, s)-generalised transformation graphs, a novel perspective based on transformation graphs

Parvez Ali; Annmaria Baby; D. Antony Xavier; Theertha Nair A; Haidar Ali; Syed Ajaz K. Kirmani

doi:10.1515/mgmc-2024-0024

Article Open Access

Study on (r, s)-generalised transformation graphs, a novel perspective based on transformation graphs

Parvez Ali , Annmaria Baby , D. Antony Xavier , Theertha Nair A , Haidar Ali and Syed Ajaz K. Kirmani

Published/Copyright: July 8, 2025

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From the journal Main Group Metal Chemistry Volume 48 Issue 1

Abstract

For a graph Q = ( V , E ) , the transformation graph are defined as graphs with vertex set being V ( Q ) ∪ E ( Q ) and edge set is described following certain conditions. In comparison with the structural descriptor of the original graph Q , the topological descriptor of its transformation graphs displays distinct characteristics related to the structure. Thus, a compound’s transformation graph descriptors can be used to model a variety of structural features of the underlying molecular structure and initiate a structural analysis. In this work, the concept of transformation graphs is extended giving rise to a novel class of graphs, the ( r , s ) -generalised transformation graphs, whose vertex set is union of r copies of V ( Q ) , and s copies of E ( Q ) , where r , s ∈ N , and the edge set are defined under certain conditions. Furthermore, this class of graphs is analysed with the help of first Zagreb index. Mainly, there are eight transformation graphs based on the criteria for edge set, but under the concept of ( r , s ) -generalised transformation graphs, infinite number of graphs can be described and analysed.

Keywords: transformation graphs; generalised transformations; molecular structure; topological descriptor; structural analysis

MSC 2010: 05C09; 05C90; 05C92; 92E10

1 Introduction

In theoretical chemistry, chemical compounds are visualised as molecular graphs with edges acting in for chemical bonds and vertices for atoms. This concept known as chemical graph theory provides a link between chemistry and mathematics, redirecting to a quantitative study of chemical compounds through graph invariants known as topological descriptors. Topological descriptors are quantitative attributes of a graph that remain invariant under graph isomorphism. It helps in a highly appealing way to identify the physical, chemical, biological, or pharmacological aspects of a chemical structure. It has broad applications in many different domains, including biology, informatics, and chemistry [1,2,3,4,5,6,7,8,9]. Interest has been sparked by the use of topological descriptors in quantitative analyses of the relationship between structure–activity and structure–property in chemistry [10,11].

Topological descriptors help in predicting various properties of a structure mathematically avoiding the experimental procedures. The vertex degree is the base for many of these descriptors [12]. In 1947, Wiener developed the Wiener index, the first known distance-based index [7]. Using the concept, Wiener defined the boiling points for alkanes. Later, in 1972, the Zagreb degree-based topological descriptors were introduced by Gutman and Trinajstić [2]. Much attention has been paid to degree-based topological descriptors, particularly Zagreb indices due to their wide applications. Furthermore, many modified forms and variants of Zagreb indices and other descriptors were also introduced. To improve the quantitative structure–activity–property relationships studies, Vukičević and Gašperov [13] established a new class of structural descriptors, called as “discrete Adriatic indices,” and it comprises 148 descriptors. For further details, see the study of Borovićanin et al. [14] and other references cited therein. Their mathematical features are detailed in the survey [15]. The applications of these topological descriptors in various fields are remarkable and researchers are coming up with new descriptors that correlate with the structural properties of a chemical structure more precisely improving its structural analysis.

Numerous graph transformations can be used to convert simpler graphs into distinct and specific graphs of chemical interest. Hence, it is important to study how different topological descriptors of such transformation are related to the corresponding descriptors of the original graph. Since a transformation graph converts the information from the original graph into new transformed structure, so if it is possible to find out the given graph from the transformed graph, then such operation may be used to figure out structural properties of the original graph considering the transformation graphs (for details, see the study of Wu and Meng [16]). A transformation graph’s topological descriptor must reflect different structural features than that of the original graph, and with the help of the same class of descriptors, a wide range of structural properties of the underlying molecules could be modelled. Various graph transformations have been defined and studied in different mathematical literature. The foundation stone for transformation graphs was laid back in 1966 by Behzad [17,18] with the introduction of the idea of total graphs. Later, in 1973, Sampathkumar and Chikkodimath [19] came up with the idea of semi total-point graph and semi total-line graph. Thus, the concept of total transformation graphs was developed. In the study by Wu and Meng [16], some new graphical transformations generalising the concept of total graph were developed by researchers. There are eight different total transformation graphs depending on the adjacency of each vertex in the transformation graph. Various descriptors and properties of these transformation graphs were studied by researchers. For further study, refer to the literature [20,21,22,23,24,25,26,27,28,29,30]. Refer previous studies [31,32,33,34,35,36] for applications of such concepts.

A transformation graph Q u v w of a graph Q is developed in such a way that their vertex set V ( Q u v w ) is V ( Q ) ∪ E ( Q ) and their edge set is defined under certain criteria that are as follows:

For three variables u , v , and w , the vertices α , β ∈ V ( Q u v w ) are adjacent if and only if

α , β ∈ V ( Q ) , α , β are adjacent in Q , if u = + and α , β are not adjacent in Q , if u = − .
α , β ∈ E ( Q ) , α , β are adjacent in Q , if v = + and α , β are not adjacent in Q , if v = − .
α ∈ V ( Q ) and β ∈ E ( Q ) , α , β are incident in Q , if w = + and α , β are not incident in Q , if w = − .

Eight transformation graphs can be developed for a given graph Q , based on the distinct 3-permutations of { + , − } . Additionally, it can be noted that there are four pairs of transformation graphs among the eight graphs, which are mutually complementary, i.e., one is isomorphic to the complement of its pair. A sample graph, Q along with its transformation graphs, is presented in Figure 1. Under these conditions, the transformation graphs are restricted to a count of eight. Motivated by this concept, an attempt has been initiated to describe infinite number of transformations from a given graph Q . This novel perspective leads to the concept of ( r , s ) -generalised transformation graphs. Experts in this field have introduced and studied the concept of k -generalised transformation graphs based on transformation graphs [37,38].

$Figure 1 Sample graph Q {\mathbb{Q}} and its eight transformation graphs.$

Figure 1

Sample graph Q and its eight transformation graphs.

In this work, a new concept, the ( r , s ) -generalised transformation graphs, is established extending the existing concept of transformation graphs. Furthermore, the first Zagreb index for a few ( r , s ) -generalised transformation graphs is presented and a structural analysis is initiated.

2 Mathematical concepts

Consider a graph Q ( V , E ) with ∣ V ( Q ) ∣ = n and ∣ E ( Q ) ∣ = m . For a vertex α ∈ V ( Q ) , the degree of the vertex α is denoted as d Q ( α ) , which is the count of edges incident to that vertex. Two vertices are said to be adjacent, if they are connected by an edge, and two edges are said to be adjacent, if they have a common end vertex.

The first Zagreb index [2] is defined as

(1) M 1 ( Q ) = ∑ α ∈ V ( Q ) ( d Q ( α ) ) 2 = ∑ α β ∈ E ( Q ) ( d Q ( α ) + d Q ( β ) )

The second Zagreb index [2] is defined as

(2) M 2 ( Q ) = ∑ α β ∈ E ( Q ) ( d Q ( α ) × d Q ( β ) )

The forgotten index (F-index) [39] is defined as

(3) F ( Q ) = ∑ α ∈ V ( Q ) ( d Q ( α ) ) 3 = ∑ α β ∈ E ( Q ) ( ( d Q ( α ) ) 2 + ( d Q ( β ) ) 2 )

3 ( r , s ) -generalised transformation graphs

Consider the simple connected graph Q = ( V , E ) . Take r copies of vertex set V ( Q ) , say V 1 ( Q ) , V 2 ( Q ) , … , V r ( Q ) and s copies of edge set E ( Q ) , say E 1 ( Q ) , E 2 ( Q ) , … , E s ( Q ) , where r , s ∈ N . Then, the ( r , s ) -generalised transformation graph, Q r s x y z with

x = ( x 1 , x 2 , … , x r ) ; x g ∈ { + , − } , y = ( y 1 , y 2 , … , y s ) ; y h ∈ { + , − } , and z = ( z 11 , z 12 , … , z 1 s , z 21 , z 22 , … , z 2 s , … , z r 1 , z r 2 , … , z r s ) ; z g h ∈ { + , − }

where 1 ≤ g ≤ r ; 1 ≤ h ≤ s , has vertex set ( ⋃ g = 1 r V g ) ⋃ ( ⋃ h = 1 s E h ) , and edge set is defined such that α , β ∈ V ( Q r s u v w ) are adjacent if and only if

α , β ∈ V g ( Q ) , α , β are adjacent in Q , if x g = + and α , β are not adjacent in Q , if x g = − .
α , β ∈ E h ( Q ) , α , β are adjacent in Q , if y h = + and α , β are not adjacent in Q , if y h = − .
α ∈ V g ( Q ) and β ∈ E h ( Q ) , α , β are incident in Q , if z g h = + and α , β are not incident in Q , if z g h = − .

3.1 Special notations

For the ( r , s ) -generalised transformation graph, Q r s x y z , consider the sets

x = ( x 1 , x 2 , … , x r ) , x g ∈ { + , − } , y = ( y 1 , y 2 , … , y s ) , y h ∈ { + , − } ,

z = ( z 11 , z 12 , … , z 1 s , z 21 , z 22 , … , z 2 s , … , z r 1 , z r 2 , … , z r s ) , z g h ∈ { + , − } , where 1 ≤ g ≤ r ; 1 ≤ h ≤ s .

If x g = + , for 1 ≤ g ≤ r , then the graph is denoted as Q r s + y z .
If x g = − , for 1 ≤ g ≤ r , then the graph is denoted as Q r s − y z .
If y h = + , for 1 ≤ h ≤ s , then the graph is denoted as Q r s x + z .
If y h = − , for 1 ≤ h ≤ s , then the graph is denoted as Q r s x − z .
If z g h = + , for 1 ≤ g ≤ r ; 1 ≤ h ≤ s , then the graph is denoted as Q r s x y + .
If z g h = − , for 1 ≤ g ≤ r ; 1 ≤ h ≤ s , then the graph is denoted as Q r s x y − .
If x , y , and z are such that x g = + , y h = + , z g h = + , for 1 ≤ g ≤ r ; 1 ≤ h ≤ s , then the graph is denoted as Q r s + + + .
If x , y , and z are such that x g = − , y h = − , z g h = − , for 1 ≤ g ≤ r ; 1 ≤ h ≤ s , then the graph is denoted as Q r s − − − .

3.2 Illustration

Consider a sample graph Q ( V , E ) with V ( Q ) = { 1 , 2 , 3 , 4 , 5 } and E ( Q ) = { 12 , 14 , 23 , 25 } , as given in Figure 2.

Example 1:

$Figure 2 Sample graph Q {\mathbb{Q}} .$

Figure 2

Sample graph Q .

Choose r = 3 and s = 3 . Hence, we have three copies each of V ( Q ) and E ( Q ) . Also, take x = { x 1 , x 2 , x 3 } , y = { y 1 , y 2 , y 3 } and z = { z 11 , z 12 , z 13 , z 21 , z 22 , z 23 , z 31 , z 32 , z 33 } , such that x g = + , y h = + , z g h = + ; for g , h ∈ { 1 , 2 , 3 } .

The resulting transformation graph is Q 33 + + + , as demonstrated in Figure 3.

$Figure 3 Transformation graph, Q 33 + + + {{\mathbb{Q}}}_{33}^{+++} .$

Figure 3

Transformation graph, Q 33 + + + .

Example 2

Choose r = 2 and s = 1 .

Take x = { x 1 , x 2 } = { + , − } , y = { y 1 } = { − } , and z = { z 11 , z 21 } = { + , + } .

The resulting transformation graph is Q 21 x − + , as demonstrated in Figure 4.

$Figure 4 Transformation graph, Q 21 x − + {{\mathbb{Q}}}_{21}^{x-+} , where x = { + , − } x=\left\{+,-\right\} .$

Figure 4

Transformation graph, Q 21 x − + , where x = { + , − } .

Example 3

Choose r = 2 and s = 2 .

Take, x = { x 1 , x 2 } = { − , − } , y = { y 1 , y 2 } = { − , + } , and z = { z 11 , z 12 , z 21 , z 22 } = { + , − , + , − } .

The resulting transformation graph is Q 22 − y z , as demonstrated in Figure 5.

$Figure 5 Transformation graph, Q 22 − y z {{\mathbb{Q}}}_{22}^{-yz} , where y = { − , + } y=\left\{-,+\right\} and z = { + , − , + , − } z=\left\{+,-,+,-\right\} .$

Figure 5

Transformation graph, Q 22 − y z , where y = { − , + } and z = { + , − , + , − } .

4 ( r , s ) -generalised transformation graphs Q r s x y + and Q r s x y −

For a graph Q , the transformation graph Q r s x y + has x = { x 1 , x 2 , … , x r } and y = { y 1 , y 2 , … , y s } , where, x g , y h ∈ { + , − } , but z = { z 11 , z 12 , … , z 1 s , z 21 , z 22 , … , z 2 s , … , z r 1 , z r 2 , … , z r s } is such that z g h = + , where, 1 ≤ g ≤ r and 1 ≤ h ≤ s .

Similarly, the transformation graph Q r s x y − has x = { x 1 , x 2 , … , x r } and y = { y 1 , y 2 , … , y s } , where, x g , y h ∈ { + , − } , but z = { z 11 , z 12 , … , z 1 s , z 21 , z 22 , … , z 2 s , … , z r 1 , z r 2 , … , z r s } is such that z g h = − , where 1 ≤ g ≤ r and 1 ≤ h ≤ s .

Let a ( p ) = ( a i ) 1 p and b ( q ) = ( b j ) 1 q be two finite sequences such that 1 ≤ a 1 < a 2 < … < a p ≤ r and 1 ≤ b 1 < b 2 < … < b q ≤ s .

Define x ( a ( p ) ) = { x 1 , x 2 , … , x r } , where

x g = + , if g = a i , 1 ≤ i ≤ p − , otherwise

and, define y ( b ( q ) ) = { y 1 , y 2 , … , y s } , where

y h = + , if h = b j , 1 ≤ j ≤ q − , otherwise

Also, for 1 ≤ p ≤ r , define x ( p ) = { x 1 , x 2 , … , x r } , where

x g = + , if 1 ≤ g ≤ p − , p < g ≤ r

and for 1 ≤ q ≤ s , define y ( q ) = { y 1 , y 2 , … , y s } , where

y h = + , if 1 ≤ h ≤ q − , q < h ≤ s

Similarly, define x ( a ( p ) ¯ ) = { x 1 , x 2 , … , x r } , where

x g = − , if g = a i , 1 ≤ i ≤ p + , otherwise

and define y ( b ( q ) ¯ ) = { y 1 , y 2 , … , y s } , where

y h = − , if h = b j , 1 ≤ j ≤ q + , otherwise

Also, for 1 ≤ p ≤ r , define x ( p ¯ ) = { x 1 , x 2 , … , x r } , where

x g = − , if 1 ≤ g ≤ p + , p < g ≤ r

and for 1 ≤ q ≤ s , define y ( q ¯ ) = { y 1 , y 2 , … , y s } , where

y h = − , if 1 ≤ h ≤ q + , q < h ≤ s

It is straightforward to see that

Q r s x ( a ( p ) ) y ( b ( q ) ) + ≅ Q r s x ( p ) y ( q ) + .
Q r s x ( a ( p ) ) y ( b ( q ) ) − ≅ Q r s x ( p ) y ( q ) − .
Q r s x ( a ( p ) ¯ ) y ( b ( q ) ¯ ) + ≅ Q r s x ( p ¯ ) y ( q ¯ ) + .
Q r s x ( a ( p ) ¯ ) y ( b ( q ) ¯ ) − ≅ Q r s x ( p ¯ ) y ( q ¯ ) − .

4.1 Special cases

Consider the transformation graph, Q r s x ( p ) y ( q ) + . If
- p = r and q = s , then the transformation graph is denoted as Q r s + + + .
- p = r and q = 0 , then the transformation graph is denoted as Q r s + − + .
- p = 0 and q = s ,then the transformation graph is denoted as Q r s − + + .
- p = 0 and q = 0 ,then the transformation graph is denoted as Q r s − − + .
Consider the transformation graph, Q r s x ( p ) y ( q ) − , if
- p = r and q = s , then the transformation graph is denoted as Q r s + + − .
- p = r and q = 0 , then the transformation graph is denoted as Q r s + − − .
- p = 0 and q = s , then the transformation graph is denoted as Q r s − + − .
- p = 0 and q = 0 , then the transformation graph is denoted as Q r s − − − .

4.2 First Zagreb index for Q r s x ( p ) y ( q ) +

Theorem 1

Let Q ( V , E ) be a graph with ∣ V ( Q ) ∣ = n and ∣ E ( Q ) ∣ = m . Then, for the transformation graph Q r s x ( p ) y ( q ) + , 1 ≤ p ≤ r ; 1 ≤ q ≤ s ,

M 1 ( Q r s x ( p ) y ( q ) + ) = { p ( s + 1 ) 2 + ( r − p ) ( s − 1 ) 2 + 4 q ( r − 1 ) + 2 ( q − s ) ( m + 2 r + 1 ) } M 1 ( Q ) + 2 s M 2 ( Q ) + s F ( Q ) + ( r − p ) { n ( n − 1 ) 2 + 2 m ( n − 1 ) ( s − 1 ) } + m q ( 2 r − 2 ) 2 + m ( s − q ) ( m + 2 r + 1 ) 2 .

Proof

Consider the graph Q r s x ( p ) y ( q ) + .

It is vertex set V ( Q r s x ( p ) y ( q ) + ) = ( ⋃ g = 1 r V g ) ⋃ ( ⋃ h = 1 s E h ) .

For p ; 1 ≤ p ≤ r , let

x g = + , if 1 ≤ g ≤ p − , p + 1 ≤ g ≤ r

and for q ; 1 ≤ q ≤ s , let

y h = + , if 1 ≤ h ≤ q − , q + 1 ≤ h ≤ s

Clearly, the elements in z are all “+.” Hence, x has p number of “+” and r − p number of “ − .” Similarly, y has q number of “+” and s − q number of “ − ” and z has r s elements, which are all “+.”

Under the given conditions, V ( Q r s x ( p ) y ( q ) + ) have two types of vertices, say,

α ∈ V ( Q r s x ( p ) y ( q ) + ) ∩ V g , where 1 ≤ g ≤ r .
γ = β δ ∈ V ( Q r s x ( p ) y ( q ) + ) ∩ E h , where 1 ≤ h ≤ s .

The degree of each vertex in Q r s x ( p ) y ( q ) + can be expressed as follows.

For a vertex α ∈ V ( Q r s x ( p ) y ( q ) + ) ∩ V g , where 1 ≤ g ≤ p ,

(4) d ( Q r s x ( p ) y ( q ) + ) ∩ V g ( α ) = ( s + 1 ) d Q ( α )

For a vertex α ∈ V ( Q r s x ( p ) y ( q ) + ) ∩ V g , where ( p + 1 ) ≤ g ≤ r ,

(5) d ( Q r s x ( p ) y ( q ) + ) ∩ V g ( α ) = ( s − 1 ) d Q ( α ) + n − 1

For a vertex γ = β δ ∈ V ( Q r s x ( p ) y ( q ) + ) ∩ E h , where 1 ≤ h ≤ q ,

(6) d ( Q r s x ( p ) y ( q ) + ) ∩ E h ( γ ) = d Q ( β ) + d Q ( δ ) + 2 ( r − 1 )

For a vertex γ = β δ ∈ V ( Q r s x ( p ) y ( q ) + ) ∩ E h , where ( q + 1 ) ≤ h ≤ s ,

(7) d ( Q r s x ( p ) y ( q ) + ) ∩ E h ( γ ) = m + 2 r + 1 − ( d Q ( β ) + d Q ( δ ) )

Applying these results, the first Zagreb index for Q r s x ( p ) y ( q ) + is computed as follows.

From Eq. 1,

M 1 ( Q ) = ∑ α ∈ V ( Q ) ( d Q ( α ) ) 2 .

Therefore,

M 1 ( Q r s x ( p ) y ( q ) + ) = ∑ α ∈ V ( Q r s x ( p ) y ( q ) + ) ( d Q r s x ( p ) y ( q ) + ( α ) ) 2 = ∑ α ∈ V ( Q r s x ( p ) y ( q ) + ) ∩ V g ( d Q r s x ( p ) y ( q ) + ( α ) ) 2 + ∑ γ ∈ V ( Q r s x ( p ) y ( q ) + ) ∩ E h ( d Q r s x ( p ) y ( q ) + ) 2 = p ∑ α ∈ V ( Q ) ( ( s + 1 ) d Q ( α ) ) 2 + ( r − p ) ∑ α ∈ V ( Q ) ( ( s − 1 ) d Q ( α ) + n − 1 ) 2 + q ∑ β δ ∈ E ( Q ) ( d Q ( β ) + d Q ( δ ) + 2 ( r − 1 ) ) 2 + ( s − q ) ∑ β δ E ( Q ) ( m + 2 r + 1 − ( d Q ( β ) + d Q ( δ ) ) ) 2

= p ( s + 1 ) 2 ∑ α ∈ V ( Q ) ( d Q ( α ) ) 2 + ( r − p ) ∑ α ∈ V ( Q ) ( ( ( s − 1 ) d Q ( α ) ) 2 + ( n − 1 ) 2 + 2 ( s − 1 ) ( n − 1 ) d Q ( α ) ) + q ∑ β δ ∈ E ( Q ) ( ( d Q ( β ) + d Q ( δ ) ) 2 + ( 2 ( r − 1 ) ) 2 + 4 ( r − 1 ) ( d Q ( β ) + d Q ( δ ) ) ) + ( s − q ) ∑ β δ E ( Q ) ( ( m + 2 r + 1 ) 2 + ( d Q ( β ) + d Q ( δ ) ) 2 − 2 ( m + 2 r + 1 ) ( d Q ( β ) + d Q ( δ ) ) ) = p ( s + 1 ) 2 M 1 ( Q ) + ( r − p ) ( ( s − 1 ) 2 M 1 ( Q ) + n ( n − 1 ) 2 + 2 m ( n − 1 ) ( s − 1 ) ) + q ( 2 M 2 ( Q ) + F ( Q ) + m ( 2 r − 2 ) 2 + 4 ( r − 1 ) M 1 ( Q ) ) + ( s − q ) ( m ( m + 2 r + 1 ) 2 + 2 M 2 ( Q ) + F ( Q ) − 2 ( m + 2 r + 1 ) M 1 ( Q ) )

□ = { p ( s + 1 ) 2 + ( r − p ) ( s − 1 ) 2 + 4 q ( r − 1 ) + 2 ( q − s ) ( m + 2 r + 1 ) } M 1 ( Q ) + 2 s M 2 ( Q ) + s F ( Q ) + ( r − p ) { n ( n − 1 ) 2 + 2 m ( n − 1 ) ( s − 1 ) } + m q ( 2 r − 2 ) 2 + m ( s − q ) ( m + 2 r + 1 ) 2

Corollary 2

Consider the graph Q ( V , E ) , such that ∣ V ( Q ) ∣ = n and ∣ E ( Q ) ∣ = m , then

M 1 ( Q r s + + + ) = ( r ( s 2 + 6 s + 1 ) − 4 s ) M 1 ( Q ) + 2 s M 2 ( Q ) + s F ( Q ) + 4 m s ( r − 1 ) 2 , M 1 ( Q r s + − + ) = ( r ( s + 1 ) 2 − 2 s ( m + 2 r + 1 ) ) M 1 ( Q ) + 2 s M 2 ( Q ) + s F ( Q ) + m s ( m + 2 r + 1 ) 2 M 1 ( Q r s − + + ) = ( r ( s − 1 ) 2 − 2 s ( m + 2 r + 1 ) ) M 1 ( Q ) + 2 s M 2 ( Q ) + s F ( Q ) + n r ( n − 1 ) 2 + 2 m r ( n − 1 ) ( s − 1 ) + 4 m s ( r − 1 ) 2 M 1 ( Q r s − − + ) = ( r ( s − 1 ) 2 − 4 s r − 4 s ) M 1 ( Q ) + 2 s M 2 ( Q ) + s F ( Q ) + n r ( n − 1 ) 2 + 2 m r ( n − 1 ) ( s − 1 ) + m s ( m + 2 r + 1 ) 2

4.3 First Zagreb index for Q r s x ( p ) y ( q ) −

Theorem 3

Let Q ( V , E ) be a graph with ∣ V ( Q ) ∣ = n and ∣ E ( Q ) ∣ = m . Then, for the transformation graph Q r s x ( p ) y ( q ) − , 1 ≤ p ≤ r ; 1 ≤ q ≤ s ,

M 1 ( Q r s x ( p ) y ( q ) − ) = { p ( 1 − s ) 2 + ( r − p ) ( s + 1 ) 2 + 2 q ( n r − 2 r − 2 ) + 2 ( q − s ) ( m + r ( a − 2 ) + 1 ) } M 1 ( Q ) + 2 s M 2 ( Q ) + s F ( Q ) + ( r − p ) { n ( n + s b − 1 ) 2 − 4 m ( n + s m − 1 ) ( s + 1 ) } + p s m 2 ( n s + 4 ( 1 − s ) ) + m q ( n r − 2 r − 2 ) 2 + m ( s − q ) ( m + r ( n − 2 ) + 1 ) 2

Proof

Consider the graph Q r s x ( p ) y ( q ) − . It is vertex set V ( Q r s x ( p ) y ( q ) − ) = ( ⋃ g = 1 r V g ) ⋃ ( ⋃ h = 1 s E h ) .

For p ; 1 ≤ p ≤ r and x = { x 1 , x 2 , … , x r } ,

x g = + , if 1 ≤ g ≤ p − , p + 1 ≤ h ≤ r

Also, for q ; 1 ≤ q ≤ s and y = { y 1 , y 2 , … , y s } ,

y h = + , if 1 ≤ h ≤ q − , q + 1 ≤ h ≤ s

Clearly, the elements in z are all “ − .” Hence, x has p number of “+” and r − p number of “ − .” Similarly, y has q number of “+” and s − q number of “ − ” and z has r s elements, which are all “ − .”

Under the given conditions, V ( Q r s x ( p ) y ( q ) − ) have two types of vertices, say,

α ∈ V ( Q r s x ( p ) y ( q ) − ) ∩ V g , where 1 ≤ g ≤ r .
γ = β δ ∈ V ( Q r s x ( p ) y ( q ) − ) ∩ E h , where 1 ≤ h ≤ s .

The degree of each vertex in Q r s x ( p ) y ( q ) − can be expressed as follows.

For a vertex α ∈ V ( Q r s x ( p ) y ( q ) − ) ∩ V g , where 1 ≤ g ≤ p ,

(8) d ( Q r s x ( p ) y ( q ) − ) ∩ V g ( α ) = s m + ( 1 − s ) d Q ( α )

For a vertex α ∈ V ( Q r s x ( p ) y ( q ) − ) ∩ V g , where ( p + 1 ) ≤ g ≤ r ,

(9) d ( Q r s x ( p ) y ( q ) − ) ∩ V g ( α ) = n + s m − 1 − ( s + 1 ) d Q ( α )

For a vertex γ = β δ ∈ V ( Q r s x ( p ) y ( q ) − ) ∩ E h , where 1 ≤ h ≤ q ,

(10) d ( Q r s x ( p ) y ( q ) − ) ∩ E h ( γ ) = d Q ( β ) + d Q ( δ ) + r ( n − 2 ) − 2

For a vertex γ = β δ ∈ V ( Q r s x ( p ) y ( q ) − ) ∩ E h , where ( q + 1 ) ≤ h ≤ s ,

(11) d ( Q r s x ( p ) y ( q ) − ) ∩ E h ( γ ) = m + r ( n − 2 ) + 1 − ( d Q ( β ) + d Q ( δ ) )

Applying these results, the first Zagreb index for Q r s x ( p ) y ( q ) − is computed as in Theorem 1.

□ M 1 ( Q r s x ( p ) y ( q ) − ) = ∑ α ∈ V ( Q r s x ( p ) y ( q ) − ) ( d Q r s x ( p ) y ( q ) − ( α ) ) 2 = ∑ α ∈ V ( Q r s x ( p ) y ( q ) − ) ∩ V g ( d Q r s x ( p ) y ( q ) − ( α ) ) 2 + ∑ γ ∈ V ( Q r s x ( p ) y ( q ) − ) ∩ E h ( d Q r s x ( p ) y ( q ) − ) 2 = p ∑ α ∈ V ( Q ) ( s m + ( 1 − s ) d Q ( α ) ) 2 + ( r − p ) ∑ α ∈ V ( Q ) ( n + s m − 1 − ( s + 1 ) d Q ( α ) ) 2 + q ∑ β δ ∈ E ( Q ) ( d Q ( β ) + d Q ( δ ) + r ( n − 2 ) − 2 ) 2 + ( s − q ) ∑ β δ E ( Q ) ( m + r ( n − 2 ) + 1 − ( d Q ( β ) + d Q ( δ ) ) ) 2 = { p ( 1 − s ) 2 + ( r − p ) ( s + 1 ) 2 + 2 q ( n r − 2 r − 2 ) + 2 ( q − s ) ( m + r ( n − 2 ) + 1 ) } M 1 ( Q ) + 2 s M 2 ( Q ) + s F ( Q ) + ( r − p ) { n ( n + s m − 1 ) 2 − 4 m ( n + s m − 1 ) ( s + 1 ) } + p s m 2 ( n s + 4 ( 1 − s ) ) + m q ( n r − 2 r − 2 ) 2 + m ( s − q ) ( m + r ( n − 2 ) + 1 ) 2

Corollary 4

Consider the graph Q ( V , E ) , such that ∣ V ( Q ) ∣ = n and ∣ E ( Q ) ∣ = m , then

M 1 ( Q r s + + − ) = ( r ( 1 − s ) 2 + 2 s ( n r − 2 r − 2 ) ) M 1 ( Q ) + 2 s M 2 ( Q ) + s F ( Q ) + n r m 2 s 2 + 4 r s m 2 ( 1 − s ) + s m ( n r − 2 r − 2 ) 2 , M 1 ( Q r s + − − ) = ( r ( 1 − s ) 2 − 2 s ( m + r ( n − 2 ) + 1 ) ) M 1 ( Q ) + 2 s M 2 ( Q ) + s F ( Q ) + n r m 2 s 2 + 4 r s m 2 ( 1 − s ) + s m ( m + r ( n − 2 ) + 1 ) 2

M 1 ( Q r s − + − ) = ( r ( s + 1 ) 2 + 2 s ( n r − 2 r − 2 ) ) M 1 ( Q ) + 2 s M 2 ( Q ) + s F ( Q ) + n r ( n + s m − 1 ) 2 − 4 m r ( s + 1 ) ( n + s m − 1 ) + m s ( n r − 2 r − 2 ) 2 M 1 ( Q r s − − − ) = ( r ( s + 1 ) 2 − 2 s ( m + r ( n − 2 ) + 1 ) ) M 1 ( Q ) + 2 s M 2 ( Q ) + s F ( Q ) + n r ( n + s m − 1 ) 2 − 4 m r ( n + m s − 1 ) ( s + 1 ) + m s ( m + r ( n − 2 ) + 1 ) 2

5 Discussion

Choose a sample graph, say P 3 , which is a path consisting of three vertices. Consider its transformation graphs ( P 3 ) r s + + + . The first Zagreb index of ( P 3 ) r s + + + for few finite values of the parameters r and s was computed and plotted graphically, and Figure 6 illustrates the graphical representation. Clearly, the bar plot increases gradually with the increase in values of parameters r and s .

$Figure 6 First Zagreb index for transformation graph, ( P 3 ) r s + + + {\left({P}_{3})}_{rs}^{+++} .$

Figure 6

First Zagreb index for transformation graph, ( P 3 ) r s + + + .

Furthermore, a scatter plotting is derived using the computed values of the first Zagreb index of ( P 3 ) r s + + + for few finite values of the parameters r and s , as given in Figure 7. The scatter diagram illustrates a few regression equations, which help determine an approximation of first Zagreb index for the graph ( P 3 ) r s + + + .

$Figure 7 Scatter diagram of first Zagreb index for transformation graph, ( P 3 ) r s + + + {\left({P}_{3})}_{rs}^{+++} .$

Figure 7

Scatter diagram of first Zagreb index for transformation graph, ( P 3 ) r s + + + .

6 Conclusion

In this work, the existing concept of generalised transformation graphs is elevated to next level by taking union of r copies of vertex set and s copies of edge set of a graph Q as the vertex set of the transformed graph giving rise to ( r , s ) -generalised transformation graphs. Under this concept, rather than the eight transformations, we can generate infinite number of transformations for a given graph. Furthermore, the first Zagreb index for few type of ( r , s ) -generalised transformation graphs was determined and analysed. This study can be expanded further by determining other topological descriptors, and structural analysis could be forwarded.

Acknowledgments

The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2025).

Funding information: The research has been financialy supoorted by the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2025).
Author contributions: Parvez Ali: writing – review, project administration; Annmaria Baby: writing – original draft, writing – review and editing, methodology, visualization, resources; D. Antony Xavier: writing – review and editing, resources, formal analysis, project administration; Theertha Nair A: writing – review and editing, resources; Haidar Ali: writing – review, formal analysis, project administration; Syed Ajaz K. Kirmani: writing – review, project administration.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2024-09-24

Accepted: 2025-05-01

Published Online: 2025-07-08

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/mgmc-2024-0024

Keywords for this article

transformation graphs; generalised transformations; molecular structure; topological descriptor; structural analysis

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