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Article

Modified Entire Forgotten Topological Index of Graphs: A Theoretical and Applied Perspective

Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23218, Saudi Arabia
*
Author to whom correspondence should be addressed.
Symmetry 2025, 17(2), 236; https://doi.org/10.3390/sym17020236
Submission received: 13 December 2024 / Revised: 13 January 2025 / Accepted: 31 January 2025 / Published: 6 February 2025
(This article belongs to the Special Issue Symmetry in Graph Algorithms and Graph Theory III)

Abstract

:
Topological indices are numerical invariants derived from graph structures that are essential tools used in computational chemistry and biology for encoding molecular information. By exploiting the inherent symmetries of molecular graphs, we develop efficient algorithms to compute these indices, particularly for large and complex molecules. These indices are rooted in vertex degrees, edge degrees, and other graph parameters, have been extensively studied, and are crucial for understanding the relationship between molecular structure and properties. Recent research has focused on the entire Zagreb indices, which integrate both vertex and edge degrees considering adjacency and incidence relationships. This paper introduces a novel variant, namely, the modified forgotten entire Zagreb index. The efficacy of this new index is underscored by its robust correlation with the physical and chemical properties of octane isomers and lower benzenoid hydrocarbons. Additionally, we derive explicit formulas for this index for several significant graph families.

1. Introduction

A graph is a mathematical structure comprising a non-empty set of vertices V and a set of edges E connecting pairs of vertices. In this study, we exclusively consider finite and undirected simple graphs. The vertex and edge sets constitute the graph’s elements. The degree of an element x (vertex or edge) is denoted by d x . If x is a vertex, then d x represents the number of edges incident to x. If x is an edge, then d x represents the number of edges adjacent to x (i.e., the number of edges sharing a common vertex with x). The order and size of a graph G are defined as the number of vertices and number of edges, respectively. A graph G is called k-regular if every vertex in G is incident to exactly k edges. The line graph L ( G ) of a graph G is a new graph that models the adjacency relationships between the edges of G. The vertex set of L ( G ) corresponds to the edge set of G, and two vertices in L ( G ) are adjacent if and only if the corresponding edges in G share a common vertex (adjacent).
For a comprehensive overview of the notations and definitions employed in this work, consult [1,2]. Path, cycle, and complete graphs with n vertices are known as P n , C n , and K n , respectively. A bipartite graph is a graph with a vertex set that can be partitioned into two disjoint and independent sets, often denoted by X and Y, such that no two vertices within the same set are adjacent. A complete bipartite graph is a bipartite graph in which every vertex in set X is adjacent to every vertex in set Y, and is denoted by K r , s , where | X | = r and | Y | = s . The Cartesian product of two graphs G 1 and G 2 , denoted by G 1 G 2 , has vertex set V ( G ) = V ( G 1 ) × V ( G 2 ) ; two distinct vertices ( a , b ) and ( c , d ) of G 1 G 2 are adjacent if either a = c and b d E ( G 2 ) or b = d and a c E ( G 1 ) . The join of two graphs G and H, denoted G + H , is a graph operation where every vertex of G is connected to every vertex of H. A wheel W n is the join of C n and K 1 . A helm graph, denoted by H n , is a graph with 2 n + 1 vertices obtained from a wheel graph W n by adjoining a pendant edge at each vertex of the cycle. A gear graph is a wheel graph with a vertex added between each pair of adjacent graph vertices of the outer cycle. A tadpole graph T r , s is obtained by joining a cycle C r and a path P s by a bridge, where r , s 3 . A book graph B m is defined as the graph Cartesian product S m P 2 . A stacked book graph B a , b is defined as the graph Cartesian product S a P b , where S a is a star graph and P b is the path graph.
Topological indices are numerical representations derived from molecular graphs. These indices capture the structural features of molecules, with atoms represented as vertices and bonds as edges. The underlying symmetry of these molecular graphs plays a crucial role in determining their topological properties. By exploiting the inherent symmetries, efficient algorithms can be developed to calculate these indices and correlate them with various physicochemical properties. These indices are valuable tools for predicting the properties of novel compounds and optimizing drug design for specific desired characteristics [3]. By understanding the relationship between molecular structure and properties, researchers can accelerate the discovery and development of new materials and pharmaceuticals. Topological indices date back to 1947, when a scientist named Wiener created the first topological index, known as the Wiener index [4], to search for boiling points. The Wiener index is defined as W ( G ) = u , v V ( G ) d ( u , v ) , where V ( G ) is the set of vertices in graph G and d ( u , v ) is the shortest path distance between vertices u and v. In 1972, Gutman and Trinajstić [5] introduced the first and second Zagreb indices. These degree-based topological indices were utilized to determine the total π -electron energy of molecules, and are defined as follows:
M 1 ( G ) = x V ( G ) d x 2 ,
M 2 ( G ) = x y E ( G ) d x d y .
The first general Zagreb index, introduced in [6], is defined as
M α ( G ) = x V ( G ) d x α ,
where α is a real number that is not equal to zero or one. In [7], a specific case of this index was investigated. This particular case is denoted as Y ( G ) and defined as Y ( G ) = x V ( G ) d x 4 .
Inspired by these advancements, numerous mathematicians and chemists have developed a wide range of topological indices to explore the diverse chemical properties of molecular graphs (structures); see [8,9,10,11,12,13,14,15,16,17,18]. Among distance-based topological indices, the Gutman index and degree distance index are particularly prominent and widely applied [19,20]. The forgotten topological index F ( G ) is defined in [21] as the sum of the cubes of the degrees of the vertices of a graph G, meaning that
F ( G ) = x V ( G ) d x 3 = x y E ( G ) d x 2 + d y 2 .
In [21], Furtula and Gutman observed that the predictive power of the forgotten index is comparable to that of the first Zagreb index, particularly for acentric factor and entropy, with both indices achieving correlation coefficients exceeding 0.95 . This suggests that the the forgotten index could be a valuable tool for assessing the chemical and pharmacological properties of drug molecules. In [22], Sun et al. explored the fundamental properties of the forgotten index and demonstrated its potential to complement the predictive capabilities of the Zagreb indices regarding physicochemical properties. More recently, Gao et al. [23] applied the forgotten index to analyze the structural features of significant drug molecules.
In 2018, Alwardi, A. et al. [24] introduced the first and second entire Zagreb topological index, as follows:
M 1 E ( G ) = x V E d x 2 ,
M 2 E ( G ) = x a d j a c e n t t o y o r x i n c i d e n t t o y d x d y .
Note that in the expressions for the first and second entire Zagreb indices, the variable x can correspond to either a vertex or an edge. When x represents a vertex, d x signifies the degree of that vertex in the graph G. Conversely, when x denotes an edge, d x refers to the degree of x in the line graph L ( G ) . The same interpretation applies to the variable y. The entire Zagreb indices are receiving a great deal of attention from many authors; for example, see [25,26,27,28,29,30,31,32,33,34].
In [33], the forgotten index was the subject of an initial study in which it was defined as
F E ( G ) = x V E d x 3 .
In this research, we introduce a novel topological index called the modified entire forgotten index, which incorporates both vertex and edge contributions to provide a more comprehensive representation of molecular structure. This index is computationally efficient even for large and complex molecules. Furthermore, we demonstrate its strong correlation with various physicochemical properties, surpassing the performance of both the forgotten index and entire forgotten index. This potential makes it a valuable tool for predicting molecular properties and aiding in the design of new molecules. We anticipate that this index will find widespread applications in fields such as chemistry, biology, and materials science, including drug design, protein structure prediction, and materials development

2. Entire Forgotten and Entire Modified Forgotten Zagreb Indices for Some Standard Graphs

This section formally introduces our novel graph invariant called the modified entire forgotten Zagreb index. We subsequently establish explicit formulas for its computation on significant classes of graphs alongside the entire forgotten Zagreb index. Throughout the proofs of the theorems and propositions, we utilize the notation E d u , d e to denote the set of vertices v incident to edge e, where d u and d e represent the degrees of vertex u and edge e, respectively, with the condition that u is incident to e.
Definition 1.
For any graph G, the modified entire forgotten Zagreb index is denoted by M F E ( G ) and defined as
M F E ( G ) = x a d j a c e n t t o y o r x i n c i d e n t t o y ( d x 2 + d y 2 ) = u v E ( G ) ( d u 2 + d v 2 ) + e f E ( L ( G ) ( d e 2 + d f 2 ) + u i n c i d e n t e ( d u 2 + d e 2 ) .
Analogously, we define the modified entire forgotten Zagreb co-index as follows:
M F E ¯ ( G ) & = u v E ( G ) ( d u 2 + d v 2 ) + e f E ( L ( G ) ) ( d e 2 + d f 2 ) + u i s n o t i n c i d e n t t o e ( d u 2 + d e 2 ) .
Proposition 1.
Let G be a graph with n vertices, and suppose that it is a k regular graph.
Proof. 
Consider a k-regular graph G with n vertices. In this case, graph G contains n k 2 edges. It is worth noting that all edges have the same degree equal to 2 k 2 ; consequently, after performing straightforward calculations, we obtain
F E ( G ) = u v E ( G ) ( d u 2 + d v 2 ) + e f E ( L ( G ) ( d e 2 + d f 2 ) = n k 4 k 3 11 k 2 + 12 k 4 .
Similarly,
M F E ( G ) = u v E ( G ) ( d u 2 + d v 2 ) + e f E ( L ( G ) ( d e 2 + d f 2 ) + u i n c i d e n t e ( d u 2 + d e 2 ) = 2 n k 2 2 k 2 3 k + 2 .
Corollary 1.
For the complete graph  K n and the cycle graph  C n , we have
i 
F E ( K n ) = 2 n n 1 2 2 n 2 7 n + 7 ,
ii 
M F E ( K n ) = 2 n n 1 2 2 n 2 7 n + 7 ,
iii 
F E ( C n ) = 16 n ,
iv 
M F E ( C n ) = 32 n .
Proposition 2.
For any path P n with n 4 vertices, we have
i 
F E ( P n ) = 16 n 36 ,
ii 
M F E ( P n ) = 32 n 70 .
Proof. 
We label the vertices and edges of the path from left to right, as follows: V ( P n ) = v 1 , v 2 , , v n and E ( P n ) = e 1 , e 2 , , e n 1 . Then, the degrees of all elements of P n are equal to 2 except for the elements v 1 , e 1 , v n , e n 1 , which all have degree 1. Thus, M F E ( P n ) = 32 n 70 and F E ( P n ) = 16 n 36 . □
Proposition 3.
Let G K a , b be a complete bipartite graph of order p = a + b and size q = a b . Then,
i 
F E ( G ) = q p 2 2 q + q p 2 3 ,
ii 
M F E ( G ) = q p 3 2 p 2 4 q + 4 p .
Proof. 
Let G K a , b be a complete bipartite graph. There are a vertices of degree b and b vertices of degree a. There are also a b edges of degree ( a + b 2 ) . Therefore,
F E ( G ) = u v E ( G ) ( d u 2 + d v 2 ) + e f E ( L ( G ) ( d e 2 + d f 2 ) = a b 3 + b a 3 + a b ( a + b 2 ) 3 .
By simple calculation, we obtain
F E ( G ) = u v E ( G ) ( d u 2 + d v 2 ) + e f E ( L ( G ) ( d e 2 + d f 2 ) = q ( p 2 2 q ) + q ( p 2 ) 3 .
For the entire forgotten index,
M F E ( G ) = u v E ( G ) ( d u 2 + d v 2 ) + e f E ( L ( G ) ( d e 2 + d f 2 ) + u i n c i d e n t e ( d u 2 + d e 2 ) = F E ( G ) + a b ( a 2 + ( a + b 2 ) 2 ) + b a ( b 2 + ( a + b 2 ) 2 ) = q p 3 2 p 2 4 q + 4 p .
Proposition 4.
For a wheel graph G W a of a + 1 vertices, we have
i 
F E ( G ) = a 4 + 4 a 3 + 3 a 2 + 92 a ,
ii 
M F E ( G ) = a 4 + 7 a 3 + 7 a 2 + 153 a .
Proof. 
Let G W a be a wheel graph. There are a vertices of degree 3, one vertex of degree a, a edges of degree four, and a edges of degree ( a + 1 ) . Thus,
F E ( G ) = x V E d x 3 = 27 a + a 3 + 64 a + a ( a + 1 ) 3 = a 4 + 4 a 3 + 3 a 2 + 92 a .
For the modified entire forgotten index,
M F E ( G ) = F E ( G ) + u i n c i d e n t e ( d u 2 + d e 2 ) .
By applying the partition of the vertices incident with the edges, as in Table 1, we obtain
M F E ( G ) = F E ( G ) + u i n c i d e n t e ( d u 2 + d e 2 ) = a 4 + 7 a 3 + 7 a 2 + 153 a .
Thanks to their advantageous properties, wheel graphs and their generalizations find applications across diverse domains such as wireless sensor networks and network vulnerability analysis. Notably, from the perspective of the central hub vertex, all other vertices and edges are located within its immediate one-hop neighborhood. This proximity simplifies information dissemination and control within the network, making it valuable for modeling various network structures.
The m-wheel graph, denoted by W n , m , is a graph constructed by connecting m copies of a cycle graph C n to a single central vertex v. Each vertex in each cycle is adjacent to the central vertex.
Theorem 1.
Let G W n , m be an m-level wheel graph. Then,
i 
F E ( G ) = m n m 3 n 3 + 4 m 2 n 2 + 3 m n + 92 ,
ii 
M F E ( G ) = m n m 3 n 3 + 7 m 2 n 2 + 7 m n + 153 .
Proof. 
Let G be an m-level graph, as in Figure 1. Obviously, all the vertices are of degree 3 except the center vertex, which is of degree m n . Thus,
F E ( G ) = m n m 3 n 3 + 4 m 2 n 2 + 3 m n + 92 .
By using the entire forgotten formula for the m-level wheel graph and the partition in Table 2, we obtain
M F E ( G ) = m n m 3 n 3 + 7 m 2 n 2 + 7 m n + 153 .
Theorem 2.
Let G be a helm graph H a . Then,
i 
F E ( G ) = a 4 + 7 a 3 + 12 a 2 + 316 a ,
ii 
M F E ( G ) = a 4 + 10 a 3 + 20 a 2 + 479 a .
Proof. 
Let G be the helm graph H a . There are a vertices of degree one, a vertices of degree four, and the center vertex with degree a. For the edges, there are a edges of degree 3, a edges of degree 6, and a edges of degree a + 2 .
Therefore, F E = a 4 + 7 a 3 + 12 a 2 + 316 a .
To obtain the formula of the modified entire forgotten index, we have
M F E ( G ) = F E ( G ) + u i n c i d e n t e ( d u 2 + d e 2 ) .
Using the partition in Table 3, we obtain
M F E ( G ) = a 4 + 10 a 3 + 20 a 2 + 479 a .
Proposition 5.
For the gear graph G G p , we obtain
i 
F E ( G ) = p p + 6 p 2 2 p + 15 ,
ii 
M F E ( G ) = p p 3 + 7 p 2 + 7 p + 163 .
Proof. 
For the gear graph, there are 2 p edges of degree 3 and p edges of degree ( p + 1 ) . In addition, there are p vertices of degree 2, p vertices of degree 3, and one vertex of degree p.
Therefore, F E ( G ) = p p + 6 p 2 2 p + 15 .
To obtain the modified forgotten index, we can use the result of the entire forgotten index and apply the vertex edge partitions in Table 4, obtaining the following result:
M F E ( G ) = p p 3 + 7 p 2 + 7 p + 163 .
Theorem 3.
Let G be a ( p , q ) -kite graph with p , q 3 . Then,
i 
F E ( G ) = 16 ( p + q ) + 62 ,
ii 
M F E ( G ) = 32 ( p + q ) + 98 .
Proof. 
For the ( p , q ) -kite graph, there are p + q 2 vertices of degree 2, one vertex of degree 3, and one vertex of degree one. In addition, there are p + q 4 edges of degree 2, one edge of degree one, and 3 edges of degree 3. Thus,
F E ( G ) = 16 ( p + q ) + 62 .
In addition to the result of the entire forgotten index, using Table 5 we obtain M F E ( G ) = 32 ( p + q ) + 98 .
Let G 1 , G 2 , G m be a set of finite pairwise disjoint graphs with v i V ( G i ) . The bridge graph B ( G 1 , G 2 , , G m ) = B ( G 1 , G 2 , , G m ; w 1 , w 2 , w d ) of { G i } i = 1 m with respect to the vertices { w i } i = 1 m is the graph obtained from graphs G 1 , , G m by connecting vertices w i and w i + 1 by an edge for all i = 1 , 2 , , m 1 ; see Figure 2.
Theorem 4.
Let G be a bridge graph over path P n , n 3 . Then,
i 
F E ( G ) = 2 8 n m + 40 m 107 ,
ii 
M F E ( G ) = 4 104 n m 258 m + 109 .
Proof. 
Let G be a bridge graph over path P n with n 3 , as in Figure 2. There are m vertices of degree one, ( m 2 ) vertices of degree three, and 2 n 2 + ( m 2 ) ( n 2 ) vertices of degree two. For the edges, there are m edges of degree one, m 3 of degree four, m edges of degree three, and 2 n 4 + ( n 3 ) ( m 2 ) of degree two. Therefore,
F E ( G ) = 2 8 n m + 40 m 107 .
By applying the partition in Table 6 and the value of entire forgotten index, we obtain
M F E ( G ) = 4 104 n m 258 m + 109 .
Theorem 5.
Let G be the m-bridge graph on cycle C n , n , m 3 . Then,
i 
F E ( G ) = 384 m + 16 m n 620 ,
ii 
M F E ( G ) = 560 m + 32 m n 866 .
Proof. 
As G is the m-bridge over a cycle C n , there are 2 vertices of degree 3, ( m 2 ) vertices of degree four, and m ( n 1 ) vertices of degree two. In addition, there are two edges of degree five, m 3 edges of degree six, four edges of degree three, ( 2 m 4 ) edges of degree four, and ( m n 2 m ) edges of degree two. Therefore,
F E ( G ) = 384 m + 16 m n 620 .
The vertex edge partition can be written as shown in Table 7.
By applying the partition in Table 7 and the value of entire forgotten index we obtain
M F E ( G ) = 560 m + 32 m n 866 .
Theorem 6.
Let G be the m-bridge graph on a complete graph K n , where n , m 3 . Then,
i 
F E ( G ) = 4 n 5 m 27 n 4 m + 113 n 3 m 56 n 3 170 n 2 m + 60 n 2 + 217 n m 244 n 46 m + 12 ,
ii 
M F E ( G ) = 4 n 5 m 22 n 4 m + 90 n 3 m 56 n 3 105 n 2 m + 30 n 2 + 160 n m 230 n 20 m 6 .
Proof. 
Let G be the m-bridge over a complete graph K n . The graph G then has n m vertices and m n 2 m n + 2 m 2 2 edges with the following degrees: 2 vertices of degree n, ( m 2 ) vertices of degree n + 2 , and m ( n 1 ) of degree n 1 . For the edges, there are two edges of degree 2 n , m 3 edges of degree 2 n + 2 , 2 n 2 edges of degree 2 n 3 , ( n 1 ) ( m 2 ) edges of degree 2 n 1 , and m ( n 1 ) ( n 2 ) / 2 edges of degree 2n-4 2 n 4 . Therefore,
F E ( G ) = 4 n 5 m 27 n 4 m + 113 n 3 m 56 n 3 170 n 2 m + 60 n 2 + 217 n m 244 n 46 m + 12 .
The vertex edge partition can be written as in Table 8.
By applying the partition in Table 8 and the value of entire forgotten index, we obtain
M F E ( G ) & = 4 n 5 m 27 n 4 m + 113 n 3 m 56 n 3 170 n 2 m + 60 n 2 + 217 n m 244 n 46 m + 12 + 2 n 2 + 2 n 1 2 + n 2 + 2 n 3 2 2 n 2 + 2 n + 1 2 + 2 n 1 2 + n + 1 2 + 4 n 2 2 m 6 + n + 1 2 + ( 2 n 2 ) 2 ( m 2 ) ( n 1 ) + n 1 2 + ( 2 n 2 ) 2 ( m 2 ) ( n 1 ) + n 1 2 + ( 2 n 3 ) 2 ( 2 n 2 ) + m n 1 2 + ( 2 n 4 ) 2 ( n 1 ) ( n 2 ) = 4 n 5 m 22 n 4 m + 90 n 3 m 56 n 3 105 n 2 m + 30 n 2 + 160 n m 230 n 20 m 6 .
A graph which has been derived from a graph G by a sequence of edge subdivision operations is called a subdivision graph of G, and is denoted by S ( G ) .
Theorem 7.
Let G be any graph of n vertices and m edges, and let H be the subdivision of G. Then,
i 
F E ( H ) = F ( G ) + Y ( G ) + 8 m ,
ii 
M F E ( H ) = 4 F ( G ) + Y ( G ) + 16 m .
Proof. 
Let G be any graph of m edges and let H be the subdivision of G. Then,
F ϵ ( H ) = x V ( H ) E ( H ) d x 3 = v V ( H ) d v 3 + e E ( H ) d e 3 = u v E ( H ) ( d u 2 + d v 2 ) + e f E ( L ( H ) ) ( d e 2 + d f 2 ) .
Through a rigorous analysis of the subdivision graph definition and subsequent calculations, we determined that
= v V ( G ) d v / G d v / G 2 + 4 + v V ( G ) d v / G 2 d v / G d v / G 2 + u v E ( G ) ( d u / G 2 + d v / G 2 ) = F ( G ) + Y ( G ) + 8 m .
To determine the formula of the modified entire forgotten index, we have
M F E ( H ) = F E ( H ) + w i n c i d e n t e ( d w / H 2 + d e / H 2 ) = F E ( H ) + v V ( G ) d v / G ( 2 d v / G 2 ) + v u E ( G ) ( d v / G 2 + 4 + d u / G 2 + 4 ) = F E ( H ) + 3 F ( G ) + 8 m .
Hence, M F E ( H ) = 4 F ( G ) + Y ( G ) + 16 m .
Theorem 8.
Let G be a graph of n vertices and m edges, and let H be its central graph. Then,
i 
F E ( H ) = 70 m 27 n 4 + 69 n 3 77 n 2 + 31 n 6 m n 3 + 42 m n 2 90 m n + 4 n 5 ,
ii 
M F E ( H ) = 50 m 22 n 4 + 46 n 3 42 n 2 + 14 n 6 m n 3 + 38 m n 2 66 m n + 4 n 5 .
Proof. 
Consider that G is a graph of n vertices and m edges with H as its central graph. Then, there are two type of vertices in H, namely, m vertices of degree two and n vertices of degree ( n 1 ) . In addition, there are 2 m edges of degree n 1 and 1 2 v V ( G ) ( n d v / G 1 ) edges of degree ( 2 n 4 ) . Thus,
F E ( H ) = 8 m + n ( n 1 ) 3 + 2 m ( n 1 ) 3 + 1 2 ( 2 n 4 ) 3 v V ( G ) ( n d v / G 1 ) = 8 m + n ( n 1 ) 3 + 2 m ( n 1 ) 3 + 4 ( n 2 ) 3 ( n 2 2 m n ) = 70 m 27 n 4 + 69 n 3 77 n 2 + 31 n 6 m n 3 + 42 m n 2 90 m n + 4 n 5 .
To obtain the expression of the modified entire forgotten index, we have
M F E ( H ) = F E ( H ) + w i n c i d e n t e ( d w / H 2 + d e / H 2 ) = F E ( H ) + ( 8 + 2 ( n 1 ) 2 ) m + 2 ( n 1 ) 2 v V ( G ) d v / G + ( ( n 1 ) 2 + ( 2 n 4 ) 2 ) v V ( G ) ( n d ( v / G ) 1 ) = F E ( H ) + ( 8 + 2 ( n 1 ) 2 ) m + 4 m ( n 1 ) 2 + ( n 2 2 m n ) ( ( n 1 ) 2 + ( 2 n 4 ) 2 ) = F E ( H ) + 5 n 4 23 n 3 + 35 n 2 17 n 4 n 2 m + 24 n m 20 m .
Hence,
M F E ( B m ) = 50 m 22 n 4 + 46 n 3 42 n 2 + 14 n 6 m n 3 + 38 m n 2 66 m n + 4 n 5 .
Theorem 9.
For any book graph B m , where m 3 ,
i 
F E ( B m ) = 2 m 4 + 8 m 3 24 m 2 + 48 m 32 .
ii 
M F E ( B m ) = 2 m 4 + 16 m 3 22 m 2 + 56 m 48 .
Proof. 
Obviously, from the definition of the book graph B m , there are 2 m 2 vertices with degree 2, two vertices of degree m, ( 2 m 2 ) edges of degree m, one edge of degree ( 2 m 2 ) , and m edges of degree ( 2 m 2 ) . Therefore,
F E ( B m ) = 2 m 4 + 8 m 3 24 m 2 + 48 m 32 .
Furthermore, employing Table 9, it can be observed that
M F E ( B m ) = 2 m 4 + 16 m 3 22 m 2 + 56 m 48 .
Theorem 10.
Let H B a , b with a , b 3 be a stacked book graph. Then,
i 
F E ( H ) = 28 a 3 + a 4 b 114 a 2 126 a + 90 a b 98 b + 14 a 3 b + 9 a 2 b + 182 ,
ii 
M F E ( H ) = 28 a 3 + a 4 b 144 a 2 246 a + 152 a b 164 b + 17 a 3 b + 26 a 2 b + 298 .
Proof. 
Let H be the stacking book graph B a , b . It is not difficult to see that the graph B a , b possesses 2 vertices of degree a, b 2 vertices of degree a + 1 , 2 a 2 vertices of degree 2, and ( a b 2 a b + 2 ) vertices of degree 3. Regarding the edges, there are 2 a 2 edges of degree a, ( a b 2 a b + 2 ) edges of degree a + 2 , 2 a 2 ) edges of degree 3, ( a b 3 a b + 3 ) edges of degree 4, two edges of degree 2 a 1 , and ( b 3 ) edges of degree 2 a . Consequently,
F E ( H ) = 2 a 3 + a + 1 3 b 2 + 8 2 a 2 + 27 a b 2 a b + 2 + a 3 2 a 2 + a + 2 3 a b 2 a b + 2 + 27 2 a 2 + 64 a b 3 a b + 3 + 8 2 a 1 3 + 8 a 3 b 3 .
Hence,
F E ( H ) = 28 a 3 + a 4 b 114 a 2 126 a + 90 a b 98 b + 14 a 3 b + 9 a 2 b + 182 .
Based on the partition of vertices incident with edges delineated in Table 10, we deduce that
M F E ( H ) = F E ( H ) + 3 2 a 2 + 4 + a 2 2 a 2 + 4 a 2 + a + 1 2 2 b 6 + 18 2 a 2 + 9 + a + 2 2 a b 2 a b + 2 + 25 2 a b 2 b 6 a + 6 + 2 a 2 + 2 a 1 2 + a + 1 2 + a + 2 2 b a b 2 a + 2 + 2 a + 1 2 + 2 a 1 2 + 2 a 2 2 a 2 .
Hence, after some calculations, we have
M F E ( H ) = 28 a 3 + a 4 b 144 a 2 246 a + 152 a b 164 b + 17 a 3 b + 26 a 2 b + 298 .
Definition 2.
[35] A firefly graph, denoted by F x , y , z , is a connected graph with a unique central vertex from which emanate x pendant edges, y pendant paths of length 2, and z triangles. The graph has n = x + 2 y + 2 z + 1 vertices and m = x + 2 y + 3 z edges, where x, y, and z are positive integers.
Theorem 11.
Consider H F x , y , z , which is a firefly graph. Then,
i 
F E ( H ) = x ( x + y + 2 z 1 ) 3 + ( y + 2 z + 1 ) ( x + y + 2 z ) 3 + x + 10 y + 24 z ,
ii 
M F E ( H ) = x ( x + y + 2 z 1 ) 3 + ( y + 2 z + 1 ) ( x + y + 2 z ) 3 + 2 x + 3 y + 6 z ( x + y + 2 z ) 2 + x x + y + 2 z 1 2 + 2 x + 21 y + 48 z .
Proof. 
The firefly graph H has the following properties: for vertices, x + y vertices of degree 1, y + 2 z vertices of degree 2, and one vertex of degree x + y + 2 z ; for edges, b edges of degree 1, z edges of degree 2, x edges of degree x + y + 2 z 1 , and y + 2 z edges of degree x + y + 2 z .
Therefore,
F E ( H ) = x + y + 8 ( y + 2 z ) + ( x + y + 2 z ) 3 + y + 8 z + x ( x + y + 2 z 1 ) 3 + ( y + 2 z ) ( x + y + 2 z ) 3
F E ( H ) = x ( x + y + 2 z 1 ) 3 + ( y + 2 z + 1 ) ( x + y + 2 z ) 3 + x + 10 y + 24 z .
From the analysis of the vertex partition incident with the edges outlined in Table 11, after some calculations we can conclude that
M F E ( H ) = F E ( H ) + x ( ( x + y + 2 z ) 2 + 1 ) + 7 y + ( y + 2 z ) ( 4 + ( x + y + 2 z ) 2 ) + 16 z + 2 ( y + 2 z ) ( x + y + 2 z ) 2 + x = F E ( H ) + 2 x + 3 y + 6 z x + y + 2 z 2 + x x + y + 2 z 1 2 + x + 11 y + 24 z = x ( x + y + 2 z 1 ) 3 + ( y + 2 z + 1 ) ( x + y + 2 z ) 3 + x + 10 y + 24 z + 2 x + 3 y + 6 z x + y + 2 z 2 + x x + y + 2 z 1 2 + x + 11 y + 24 z .

3. Utility in Predicting Molecular Properties

This section employs the Quantitative Structure–Property Relationship (QSPR) methodology to evaluate the efficacy of the newly introduced modified entire forgotten topological index in characterizing molecular structural features. We emphasize the significance of the index by investigating its correlations with a diverse array of chemical and physical properties, including the acentric factor (AF), enthalpy of vaporization (HVAP), entropy (S), heat capacity at constant pressure (CP), motor octane number (MON), density (DENS), and molar volume (MV). For more information about these chemical and physical concepts and their values, see [36,37].
The dataset corresponds to octane isomers and is derived from [36,37,38,39], as shown in Figure 3. Using SPSS software, we analyzed and determined the correlations between the modified entire forgotten topological index and the previously mentioned physicochemical properties of octane isomers. The results of these correlations are detailed in Table 12 and Table 13. By employing a linear model, we determined the correlation between the modified entire forgotten topological index and various chemical and physical properties of 18 octane isomers. The results indicate strong correlations, as presented in Table 14 and Table 15.
A comparison of the correlations between the modified entire forgotten index M F E ( G ) , the forgotten index F ( G ) , and the entire forgotten index F E ( G ) with respect to the properties of the octane isomers (acentric factor (AF), enthalpy of vaporization (HVAP), entropy (S), heat capacity at constant pressure (CP), motor octane number (MON), density (DENS), and molar volume (MV)) reveals that M F E ( G ) exhibits stronger correlations with almost all of the proprieties compared to the other two indices, as demonstrated in corr1,SYMM2. Figure 4 illustrates the correlations between the modified entire forgotten index and the physicochemical properties of octane isomers.
This finding underscores the superior predictive power of our newly introduced index in characterizing molecular properties.
We obtained the following regression equations:
S = 0.027476386 M E F ( G ) + 116.7793982 , H V A P = 0.010123825 M E F ( G ) + 73.35122234 , A F = 0.000219195 M E F ( G ) + 0.426444322 , C P = 0.00314114 M E F ( G ) + 25.09394843 , M O N = 25.0323908 M E F ( G ) 2.021300133 , D E N S = 0.000134227 M E F ( G ) + 0.660030483 , M V = 0.026899573 M E F ( G ) + 170.9916232 .
To further validate the predictive power of the new index, we investigated its correlation with the properties of benzenoid hydrocarbons. We explored the relationship between the experimentally determined boiling point (BP), π -electron energy, molecular weight (MW), polarizability (P), and molar refractivity (MR) of 21 lower benzenoid hydrocarbons. The molecular structures of these compounds are depicted in Figure 5. The forgotten, entire forgotten, and modified entire forgotten indices were calculated using established mathematical formulas. The experimental property values were sourced from [36,37,38] and are tabulated in Table 16.
The modified entire forgotten index exhibits excellent correlations with various properties, as shown in Table 17. Figure 6 illustrates the strong correlations between the modified entire forgotten index and the physicochemical properties of benzenoid hydrocarbons.
Linear regression analysis was employed to establish quantitative relationships between the modified entire forgotten index and the experimental boiling point (BP), π -electron energy, molecular weight (MW), polarizability (P), and molar refractivity (MR) of 21 lower benzenoid hydrocarbons, as follows:
B P = 0.189807331602268 [ M E F ] + 144.14744242145 , π e l e c t r o n e n e r g y = 0.0779174749698 [ M E F ] + 7.69640713153 , M W = 0.10240360571495 [ M E F ] + 110.267097355341 , P O = 0.014925077683291 [ M E F ] + 13.02714794123 , M R = 0.037638697447096 [ M E F ] + 32.841456251953 .

4. Conclusions

This study introduces the modified forgotten entire index, a novel topological descriptor that incorporates vertex and edge degrees while considering adjacency and incidence relationships within molecular graphs. By capitalizing on the inherent symmetries found in these systems, we have developed efficient algorithms for its computation, particularly for large and complex molecular structures.
The predictive power of the entire and modified entire forgotten indices was evaluated by analyzing their correlations with key physicochemical properties of octane isomers and lower benzenoid hydrocarbons. The observed strong correlations demonstrate the potential of our new index as a valuable tool for Quantitative Structure–Property Relationship (QSPR) modeling in various domains, including computational chemistry and biology. Furthermore, we have derived explicit formulas for these indices in several important graph families, providing a solid foundation for further theoretical investigations and applications.
Potential avenues for future research include extending the applicability of the entire and modified entire forgotten indices as well as investigating their predictive power for a wider range of molecular properties and biological activities, such as toxicity, drug-likeness, and protein–ligand interactions.
In addition, QSPR models can be further developed by utilizing the entire and modified entire forgotten indices as key descriptors in machine learning algorithms to build predictive models for various molecular properties.
Applying the entire and modified entire forgotten indices to specific applications could provide a new a way of exploring drug design, materials science, and other relevant fields by developing specific applications and case studies.
Finally, investigating the mathematical properties of the entire and modified entire forgotten indices could help to further explore their respective mathematical properties, such as their bounds and relationships with other graph invariants.

Author Contributions

Conceptualization, A.S. and N.Z.; methodology, N.Z. and M.S.A.; formal analysis, A.S., N.Z., and M.S.A.; investigation, A.S., N.Z., and M.S.A.; writing—original draft preparation, M.S.A.; writing—review and editing, M.S.A.; supervision, A.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The data used to support the findings of this study are available within this article.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The m-level wheel graph w n , m .
Figure 1. The m-level wheel graph w n , m .
Symmetry 17 00236 g001
Figure 2. The bridge graph over path P n .
Figure 2. The bridge graph over path P n .
Symmetry 17 00236 g002
Figure 3. Hydrogen-deleted molecular graphs of octane isomers.
Figure 3. Hydrogen-deleted molecular graphs of octane isomers.
Symmetry 17 00236 g003
Figure 4. Linear relation of modified entire forgotten index with different properties of octanes.
Figure 4. Linear relation of modified entire forgotten index with different properties of octanes.
Symmetry 17 00236 g004
Figure 5. Molecular structures of 21 lower benzenoid hydrocarbons.
Figure 5. Molecular structures of 21 lower benzenoid hydrocarbons.
Symmetry 17 00236 g005
Figure 6. Predictive power of the modified entire forgotten index, showing correlations with physicochemical properties of benzenoid hydrocarbons.
Figure 6. Predictive power of the modified entire forgotten index, showing correlations with physicochemical properties of benzenoid hydrocarbons.
Symmetry 17 00236 g006
Table 1. Partition of vertices incident with edges in W a .
Table 1. Partition of vertices incident with edges in W a .
E d v , d e , where v is incident to e Number of pairs ( d u , d e )
E 3 , 4 2 a
E 3 , a + 1 a
E a , a + 1 a
Table 2. Vertex edge partition incidence of the m-level wheel graph W n , m .
Table 2. Vertex edge partition incidence of the m-level wheel graph W n , m .
E d v , d e , where v is incident to e Number of pairs ( d u , d e )
E m n , m n + 1 m n
E 3 , m n + 1 m n
E 3 , 4 2 m n
Table 3. Partition of the vertices incident with the edges in the helm graph.
Table 3. Partition of the vertices incident with the edges in the helm graph.
E d v , d e , where v is incident to e Number of pairs in each type
E 1 , 3 a
E 4 , 3 a
E 4 , 6 2 a
E 4 , a + 2 a
E a , a + 2 a
Table 4. The vertex edge incidence partition in the gear graph.
Table 4. The vertex edge incidence partition in the gear graph.
E d v , d e , where v is incident to e Number of pairs
E p , p + 1 p
E 3 , 3 2 p
E 2 , 3 2 p
E 3 , p + 1 p
Table 5. The vertex edge partition incidence in the ( p , q ) -kite graph.
Table 5. The vertex edge partition incidence in the ( p , q ) -kite graph.
E d v , d e , where v is incident to e Number of pairs
E 1 , 1 1
E 2 , 1 1
E 2 , 2 2 p + 2 q 8
E 3 , 3 3
E 2 , 3 3
Table 6. The vertex edge partition in the bridge graph over path P n .
Table 6. The vertex edge partition in the bridge graph over path P n .
E d v , d e , where v is incident to e Number of pairs
E 2 , 2 2 m n 6 m + 4
E 2 , 1 m
E 1 , 1 m
E 2 , 3 m
E 3 , 3 m
E 3 , 4 2 m 6
Table 7. The vertex edge partition in the m-bridge over a cycle C n .
Table 7. The vertex edge partition in the m-bridge over a cycle C n .
E d v , d e , where v is incident to e Number of pairs
E 3 , 5 2
E 3 , 3 4
E 4 , 5 2
E 2 , 4 2 m 4
E 4 , 4 2 m 4
E 4 , 6 2 m 6
E 2 , 3 4
E 2 , 2 2 m n 4 m
Table 8. The vertex edge partition incidence in the m-bridge over a complete graph K n .
Table 8. The vertex edge partition incidence in the m-bridge over a complete graph K n .
E d v , d e , where v is incident to e Number of pairs
E n , 2 n 1 2
E n , 2 n 3 2 n 2
E n + 1 , 2 n 1 2
E n + 1 , 2 n 2 m 6
E n + 1 , 2 n 2 ( m 2 ) ( n 1 )
E n 1 , 2 n 2 ( m 2 ) ( n 1 )
E n 1 , 2 n 3 2 n 2
E n 1 , 2 n 4 m ( n 1 ) ( n 2 )
Table 9. The partition of the vertices incident with the edges in the book graph.
Table 9. The partition of the vertices incident with the edges in the book graph.
E d v , d e , where v is incident to e Number of pairs
E 2 , m 2 m 2
E 2 , 2 2 m 2
E m , m 2 m 2
E m , 2 m 2 2
Table 10. The partition of the vertices incident with the edges in the stacked book graph.
Table 10. The partition of the vertices incident with the edges in the stacked book graph.
E d v , d e , where v is incident to e Number of pairs
E 2 , 3 2 a 2
E 2 , a 2 a 2
E a + 1 , 2 a 2 b 6
E 3 , 3 2 a 2
E 3 , a + 2 a b 2 a b + 2
E 3 , 4 2 b a 2 b 6 a + 6
E a , 2 a 1 2
E a + 1 , a + 2 b a b 2 a + 2
E a + 1 , 2 a 1 2
E a , a 2 a 2
Table 11. The vertex partition incident with the edges in the firefly graph.
Table 11. The vertex partition incident with the edges in the firefly graph.
E d v , d e , where v is incident to e Number of pairs
E 1 , x + y + 2 z 1 x
E 1 , 1 y
E 2 , 1 y
E 2 , x + y + 2 z y + 2 z
E 2 , 2 2 z
E x + y + 2 z , x + y + 2 z y + 2 z
E x + y + 2 z , x + y + 2 z 1 x
Table 12. Experimental physico-chemical properties (S, HVAP, AF) and theoretical indices for octane isomers.
Table 12. Experimental physico-chemical properties (S, HVAP, AF) and theoretical indices for octane isomers.
OctanesFEFMEFSHVAPAF
n-Oct(1)5092186111.6773.190.397898
2-M-Hept(2)62130252109.8470.30.377916
3-M-Hept(3)62142268111.2671.30.371002
4-M-Hept(4)62142268109.3270.910.371504
3-E-Hex(5)62154284109.4371.70.362472
2,2-M-Hex(6)92254450103.4267.70.339426
2,3-M-Hex(7)74198356108.0270.20.348247
2,4-M-Hex(8)74180334106.9868.50.344223
2,5-M-Hex(9)74168318105.7268.60.35683
3,3-M-Hex(10)92192488104.7468.50.322596
3,4-M-Hex(11)74210372106.5970.20.340345
2-M-3-E-Pent(12)74210372106.0669.70.332433
3-M-3-E-Pent(13)92314526101.4869.30.306899
2,2,3-M-Pent(14)104346582101.3167.30.300816
2,2,4-M-Pent(15)104292516104.0964.870.30537
2,3,3-M-Pent(16)104364604102.0668.10.293177
2,3,4-M-Pent(17)86254444102.3968.370.317422
2,2,3,3-M-But(18)13451282693.0666.20.255294
Table 13. Experimental physico-chemical properties (CP, MON, DENS, and MV).
Table 13. Experimental physico-chemical properties (CP, MON, DENS, and MV).
OctanesCPMONDENSMV
n-Oct(1)24.64 0.7025162.6050
2-M-Hept(2)24.823.100.6980163.6530
3-M-Hept(3)25.635.000.7058161.8450
4-M-Hept(4)25.639.000.7046162.1200
3-E-Hex(5)25.7452.400.7136160.0760
2,2-M-Hex(6)25.677.400.6953164.2890
2,3-M-Hex(7)26.678.900.7121160.4130
2,4-M-Hex(8)25.869.900.7004163.0930
2,5-M-Hex(9)2555.700.6935164.7150
3,3-M-Hex(10)27.283.400.7100160.8870
3,4-M-Hex(11)27.481.700.7200158.6530
2-M-3-E-Pent(12)27.488.100.7193158.8070
3-M-3-E-Pent(13)28.988.700.7274157.0390
2,2,3-M-Pent(14)28.299.900.7161159.5170
2,2,4-M-Pent(15)25.5100.000.6919165.0960
2,3,3-M-Pent(16)2999.400.7262157.2980
2,3,4-M-Pent(17)27.695.900.7191158.8510
2,2,3,3-M-But(18)24.5 0.8242138.5980
Table 14. Correlation coefficient of F ( G ) , F E ( G ) , and M F E ( G ) with entropy (S), enthalpy of vaporization (HVAP), and acentric factor (AF).
Table 14. Correlation coefficient of F ( G ) , F E ( G ) , and M F E ( G ) with entropy (S), enthalpy of vaporization (HVAP), and acentric factor (AF).
EntropyEnthalpy of VaporizationAcentric Factor
F ( G ) −0.952722167−0.871572202−0.96504682
F E ( G ) −0.954461897−0.749627899−0.95719767
M F E ( G ) −0.961615599 2−0.789941136−0.977606035
Table 15. Correlation coefficient of F ( G ) , F E ( G ) , and M F E ( G ) with heat capacity at constant pressure (CP), motor octane number (MON), density (DENS), and molar volume (MV).
Table 15. Correlation coefficient of F ( G ) , F E ( G ) , and M F E ( G ) with heat capacity at constant pressure (CP), motor octane number (MON), density (DENS), and molar volume (MV).
Heat CapacityMotor Octane NumberDensityMolar Volume
F ( G ) 0.2741592120.8504208750.648498071−0.642118529
F E ( G ) 0.323190870.8438025770.75780613−0.75794845
M F E ( G ) 0.3455144060.8718622270.725891842−0.725790445
Table 16. Forgotten, entire forgotten, and modified entire forgotten topological indices along with experimental boiling point (BP), π π -electron energy, molecular weight (MW), polarizability (P), and molar refractivity (MR) for 21 lower benzenoid hydrocarbons.
Table 16. Forgotten, entire forgotten, and modified entire forgotten topological indices along with experimental boiling point (BP), π π -electron energy, molecular weight (MW), polarizability (P), and molar refractivity (MR) for 21 lower benzenoid hydrocarbons.
CompoundsFEFMEFBPBI-ELEMWPOMR
Naphthalene11833860821813.683128.1717.544.1
Phenanthrene188598104633819.448178.2324.661.9
Anthracene188580102434019.314178.2324.661.9
Chrysene258858148443125.192228.331.679.8
Benzo[a]anthracene258840146242525.101228.331.679.8
Triphenylene258876150642925.275228.331.679.8
Tetracene258822144044025.188228.331.679.8
Benzo[a]pyrene3121086185849628.222252.335.890.3
Benzo[e]pyrene3121104188049328.336252.335.890.3
Perylene3121104188049728.245252.335.890.3
Anthanthrene3661314223254731.253276.340100.8
Benzo[ghi]perylene3661332225454231.425276.340100.8
Dibenz[a,c]anthracene3281118192253530.942278.338.797.6
Dibenz[a,h]anthracene3281100190053530.881278.338.797.6
Dibenz[a,j]anthracene3281100190053130.88278.338.797.6
Picene3281118192251930.943278.338.797.6
Coronene4201560262859034.572300.444.1111.4
Dibenzo[a,h]pyrene3821346229659633.928302.442.9108.1
Dibenzo[a,i]pyrene3821346229659433.954302.442.9108.1
Dibenzo[a,l]pyrene3821364231859534.031302.442.9108.1
Pyrene242807140139322.506202.2528.772.5
Table 17. Correlation analysis of forgotten, entire forgotten, and modified entire forgotten topological indices with physicochemical properties of benzenoid hydrocarbons.
Table 17. Correlation analysis of forgotten, entire forgotten, and modified entire forgotten topological indices with physicochemical properties of benzenoid hydrocarbons.
BP π -Electron EnergyMWPOMR
F ( G ) 0.9866250.9878010.9793820.9928530.992853
F E ( G ) 0.9708230.9721340.9598810.9796360.979682
M F E ( G ) 0.9749550.9761750.9647390.9831640.983215
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Saleh, A.; Zeyada, N.; Alharthi, M.S. Modified Entire Forgotten Topological Index of Graphs: A Theoretical and Applied Perspective. Symmetry 2025, 17, 236. https://doi.org/10.3390/sym17020236

AMA Style

Saleh A, Zeyada N, Alharthi MS. Modified Entire Forgotten Topological Index of Graphs: A Theoretical and Applied Perspective. Symmetry. 2025; 17(2):236. https://doi.org/10.3390/sym17020236

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Saleh, Anwar, Nasr Zeyada, and Musab S. Alharthi. 2025. "Modified Entire Forgotten Topological Index of Graphs: A Theoretical and Applied Perspective" Symmetry 17, no. 2: 236. https://doi.org/10.3390/sym17020236

APA Style

Saleh, A., Zeyada, N., & Alharthi, M. S. (2025). Modified Entire Forgotten Topological Index of Graphs: A Theoretical and Applied Perspective. Symmetry, 17(2), 236. https://doi.org/10.3390/sym17020236

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