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

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 $L\left(G\right)$ of a graph G is a new graph that models the adjacency relationships between the edges of G. The vertex set of $L\left(G\right)$ corresponds to the edge set of G, and two vertices in $L\left(G\right)$ 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 $\left|X\right|=r$ and $\left|Y\right|=s$. The Cartesian product of two graphs $G_1$ and $G_2$, denoted by $G_1\square G_2$, has vertex set $V\left(G\right)=V\left(G_1\right)\times V\left(G_2\right)$; two distinct vertices $(a,b)$ and $(c,d)$ of $G_1\square G_2$ are adjacent if either $a=c$ and $bd\in E\left(G_2\right)$ or $b=d$ and $ac\in E\left(G_1\right)$. 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 $2n+1$ 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 $r,s\ge 3$. A book graph $B_m$ is defined as the graph Cartesian product $S_m\square P_2$. A stacked book graph $B_{a,b}$ is defined as the graph Cartesian product $S_a\square P_b$, 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 $\begin{array}{c}\hfill W\left(G\right)=\sum _{u,v\in V\left(G\right)}d(u,v)\end{array}$, where $V\left(G\right)$ 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 $\pi$-electron energy of molecules, and are defined as follows:

$$\begin{array}{c}\hfill M_1\left(G\right)=\sum _{x\in V\left(G\right)}{d}_x^2,\end{array}$$

$$\begin{array}{c}\hfill M_2\left(G\right)=\sum _{xy\in E\left(G\right)}{d}_x{d}_y.\end{array}$$

The first general Zagreb index, introduced in [6], is defined as

$$M_{\alpha }\left(G\right)=\sum _{x\in V\left(G\right)}{d}_x^{\alpha },$$

where $\alpha$ 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 $Y\left(G\right)$ and defined as $Y\left(G\right)={\sum }_{x\in V\left(G\right)}{d}_x^4$.

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 $F\left(G\right)$ is defined in [21] as the sum of the cubes of the degrees of the vertices of a graph G, meaning that

$$\begin{array}{c}\hfill F\left(G\right)=\sum _{x\in V\left(G\right)}{d}_x^3=\sum _{xy\in E\left(G\right)}{d}_x^2+d_y^2.\end{array}$$

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:

$$M_1^{\mathcal{E}}\left(G\right)=\sum _{x\in V\cup E}{d}_x^2,$$

$$M_2^{\mathcal{E}}\left(G\right)=\sum _{\begin{array}{c}xadjacenttoy\\ orxincidenttoy\end{array}}{d}_x{d}_y.$$

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 $L\left(G\right)$. 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

$$\begin{array}{c}\hfill F^{\mathcal{E}}\left(G\right)=\sum _{x\in V\cup E}{d}_x^3.\end{array}$$

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 ${\mathcal{E}}_{d_u,d_e}$ 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 $M{F}^{\mathcal{E}}\left(G\right)$ and defined as

$$M{F}^{\mathcal{E}}\left(G\right)=\sum _{\begin{array}{c}xadjacenttoy\\ orxincidenttoy\end{array}}(d_x^2+d_y^2)=\sum _{uv\in E\left(G\right)}(d_u^2+d_v^2)+\sum _{ef\in E\left(L\right(G)}(d_e^2+d_f^2)+\sum _{uincidente}(d_u^2+d_e^2).$$

Analogously, we define the modified entire forgotten Zagreb co-index as follows:

$$\overline{M{F}^{\mathcal{E}}}\left(G\right)\&=\sum _{\begin{array}{c}uv\notin E\left(G\right)\end{array}}(d_u^2+d_v^2)+\sum _{ef\notin E\left(L\right(G\left)\right)}(d_e^2+d_f^2)+\sum _{uisnotincidenttoe}(d_u^2+d_e^2).$$

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 ${\displaystyle \frac{nk}{2}}$ edges. It is worth noting that all edges have the same degree equal to $2k-2$; consequently, after performing straightforward calculations, we obtain

$$\begin{array}{c}\hfill F^{\mathcal{E}}\left(G\right)=\sum _{uv\in E\left(G\right)}(d_u^2+d_v^2)+\sum _{ef\in E\left(L\right(G)}(d_e^2+d_f^2)=nk\left(4k^3-11k^2+12k-4\right).\end{array}$$

Similarly,

$$M{F}^{\mathcal{E}}\left(G\right)=\sum _{uv\in E\left(G\right)}(d_u^2+d_v^2)+\sum _{ef\in E\left(L\right(G)}(d_e^2+d_f^2)+\sum _{uincidente}(d_u^2+d_e^2)=2n{k}^2\left(2k^2-3k+2\right).$$

□

Corollary 1.

For the complete graph $K_n$and the cycle graph $C_n$, we have

i 

$F^{\mathcal{E}}\left(K_n\right)=2n{\left(n-1\right)}^2\left(2n^2-7n+7\right)$,

ii 

$M{F}^{\mathcal{E}}\left(K_n\right)=2n{\left(n-1\right)}^2\left(2n^2-7n+7\right)$,

iii 

$F^{\mathcal{E}}\left(C_n\right)=16n$,

iv 

$M{F}^{\mathcal{E}}\left(C_n\right)=32n$.

Proposition 2.

For any path $P_n$ with $n\ge 4$ vertices, we have

i 

$F^{\mathcal{E}}\left(P_n\right)=16n-36$,

ii 

$M{F}^{\mathcal{E}}\left(P_n\right)=32n-70$.

Proof. 

We label the vertices and edges of the path from left to right, as follows: $V\left(P_n\right)$=$\left\{v_1,v_2,\dots ,v_n\right\}$ and $E\left(P_n\right)=\left\{e_1,e_2,\dots ,e_{n-1}\right\}$. 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, $M{F}^{\mathcal{E}}\left(P_n\right)=32n-70$ and $F^{\mathcal{E}}\left(P_n\right)=16n-36$. □

Proposition 3.

Let $G\cong K_{a,b}$ be a complete bipartite graph of order $p=a+b$ and size $q=ab$. Then,

i 

$F^{\mathcal{E}}\left(G\right)=q\left(p^2-2q\right)+q{\left(p-2\right)}^3$,

ii 

$M{F}^{\mathcal{E}}\left(G\right)=q\left(p^3-2p^2-4q+4p\right)$.

Proof. 

Let $G\cong K_{a,b}$ be a complete bipartite graph. There are a vertices of degree b and b vertices of degree a. There are also $ab$ edges of degree $(a+b-2)$. Therefore,

$$\begin{array}{c}\hfill F^{\mathcal{E}}\left(G\right)=\sum _{uv\in E\left(G\right)}(d_u^2+d_v^2)+\sum _{ef\in E\left(L\right(G)}(d_e^2+d_f^2)=a{b}^3+b{a}^3+ab{(a+b-2)}^3.\end{array}$$

By simple calculation, we obtain

$$\begin{array}{c}\hfill F^{\mathcal{E}}\left(G\right)=\sum _{uv\in E\left(G\right)}(d_u^2+d_v^2)+\sum _{ef\in E\left(L\right(G)}(d_e^2+d_f^2)=q(p^2-2q)+q{(p-2)}^3.\end{array}$$

For the entire forgotten index,

$$\begin{array}{c}\hfill M{F}^{\mathcal{E}}\left(G\right)=\sum _{uv\in E\left(G\right)}(d_u^2+d_v^2)+\sum _{ef\in E\left(L\right(G)}(d_e^2+d_f^2)+\sum _{uincidente}(d_u^2+d_e^2)\\ \hfill =F^{\mathcal{E}}\left(G\right)+ab(a^2+{(a+b-2)}^2)+ba(b^2+{(a+b-2)}^2)\\ \hfill =q\left(p^3-2p^2-4q+4p\right).\end{array}$$

□

Proposition 4.

For a wheel graph $G\cong W_a$ of $a+1$ vertices, we have

i 

$\begin{array}{c}\hfill F^{\mathcal{E}}\left(G\right)=a^4+4a^3+3a^2+92a,\end{array}$

ii 

$\begin{array}{c}\hfill M{F}^{\mathcal{E}}\left(G\right)=a^4+7a^3+7a^2+153a.\end{array}$

Proof. 

Let $G\cong W_a$ 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,

$$\begin{array}{c}\hfill F^{\mathcal{E}}\left(G\right)=\sum _{x\in V\cup E}{d}_x^3=27a+a^3+64a+a{(a+1)}^3\\ \hfill =a^4+4a^3+3a^2+92a.\end{array}$$

For the modified entire forgotten index,

$$\begin{array}{c}\hfill M{F}^{\mathcal{E}}\left(G\right)=F^{\mathcal{E}}\left(G\right)+\sum _{uincidente}(d_u^2+d_e^2).\end{array}$$

By applying the partition of the vertices incident with the edges, as in Table 1, we obtain

$$\begin{array}{c}\hfill M{F}^{\mathcal{E}}\left(G\right)=F^{\mathcal{E}}\left(G\right)+\sum _{uincidente}(d_u^2+d_e^2)=a^4+7a^3+7a^2+153a.\end{array}$$

□

Table 1. Partition of vertices incident with edges in $W_a$.

${\mathcal{E}}_{{\mathit{d}}_{\mathit{v}},{\mathit{d}}_{\mathit{e}}},\mathit{where}\mathit{v}\mathit{is}\mathit{incident}\mathit{to}\mathit{e}$	Number of pairs $({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{e}})$
${\mathcal{E}}_{3,4}$	$2a$
${\mathcal{E}}_{3,a+1}$	a
${\mathcal{E}}_{a,a+1}$	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 $G\cong W_{n,m}$ be an m-level wheel graph. Then,

i 

$\begin{array}{c}\hfill F^{\mathcal{E}}\left(G\right)=mn\left(m^3{n}^3+4m^2{n}^2+3mn+92\right),\end{array}$

ii 

$\begin{array}{c}\hfill M{F}^{\mathcal{E}}\left(G\right)=mn\left(m^3{n}^3+7m^2{n}^2+7mn+153\right).\end{array}$

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 $mn$. Thus,

$$\begin{array}{c}\hfill F^{\mathcal{E}}\left(G\right)=mn\left(m^3{n}^3+4m^2{n}^2+3mn+92\right).\end{array}$$

By using the entire forgotten formula for the m-level wheel graph and the partition in Table 2, we obtain

$$\begin{array}{c}\hfill M{F}^{\mathcal{E}}\left(G\right)=mn\left(m^3{n}^3+7m^2{n}^2+7mn+153\right)\end{array}.$$

□

Figure 1. The m-level wheel graph $w_{n,m}$.

Table 2. Vertex edge partition incidence of the m-level wheel graph $W_{n,m}$.

${\mathcal{E}}_{{\mathit{d}}_{\mathit{v}},{\mathit{d}}_{\mathit{e}}},\mathit{where}\mathit{v}\mathit{is}\mathit{incident}\mathit{to}\mathit{e}$	Number of pairs $({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{e}})$
${\mathcal{E}}_{mn,mn+1}$	$mn$
${\mathcal{E}}_{3,mn+1}$	$mn$
${\mathcal{E}}_{3,4}$	$2mn$

Theorem 2.

Let G be a helm graph $H_a$. Then,

i 

$F^{\mathcal{E}}\left(G\right)=a^4+7a^3+12a^2+316a,$

ii 

$M{F}^{\mathcal{E}}\left(G\right)=a^4+10a^3+20a^2+479a.$

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, $F^{\mathcal{E}}=a^4+7a^3+12a^2+316a.$

To obtain the formula of the modified entire forgotten index, we have

$$\begin{array}{c}\hfill M{F}^{\mathcal{E}}\left(G\right)=F^{\mathcal{E}}\left(G\right)+\sum _{uincidente}(d_u^2+d_e^2).\end{array}$$

Using the partition in Table 3, we obtain

Table 3. Partition of the vertices incident with the edges in the helm graph.

${\mathcal{E}}_{{\mathit{d}}_{\mathit{v}},{\mathit{d}}_{\mathit{e}}},\mathit{where}\mathit{v}\mathit{is}\mathit{incident}\mathit{to}\mathit{e}$	Number of pairs in each type
${\mathcal{E}}_{1,3}$	a
${\mathcal{E}}_{4,3}$	a
${\mathcal{E}}_{4,6}$	$2a$
${\mathcal{E}}_{4,a+2}$	a
${\mathcal{E}}_{a,a+2}$	a

$M{F}^{\mathcal{E}}\left(G\right)=a^4+10a^3+20a^2+479a.$ □

Proposition 5.

For the gear graph $G\cong G_p$, we obtain

i 

$F^{\mathcal{E}}\left(G\right)=p\left(p+6\right)\left(p^2-2p+15\right),$

ii 

$\begin{array}{cc}\hfill M{F}^{\mathcal{E}}\left(G\right)& =p\left(p^3+7p^2+7p+163\right).\hfill \end{array}$

Proof. 

For the gear graph, there are $2p$ 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, $F^{\mathcal{E}}\left(G\right)=p\left(p+6\right)\left(p^2-2p+15\right).$

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:

$$\begin{array}{cc}\hfill M{F}^{\mathcal{E}}\left(G\right)& =p\left(p^3+7p^2+7p+163\right).\hfill \end{array}$$

□

Table 4. The vertex edge incidence partition in the gear graph.

${\mathcal{E}}_{{\mathit{d}}_{\mathit{v}},{\mathit{d}}_{\mathit{e}}},\mathit{where}\mathit{v}\mathit{is}\mathit{incident}\mathit{to}\mathit{e}$	Number of pairs
${\mathcal{E}}_{p,p+1}$	p
${\mathcal{E}}_{3,3}$	$2p$
${\mathcal{E}}_{2,3}$	$2p$
${\mathcal{E}}_{3,p+1}$	p

Theorem 3.

Let G be a $(p,q)$-kite graph with $p,q\ge 3$. Then,

i 

$F^{\mathcal{E}}\left(G\right)=16(p+q)+62$,

ii 

$M{F}^{\mathcal{E}}\left(G\right)=32(p+q)+98.$

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,

$$F^{\mathcal{E}}\left(G\right)=16(p+q)+62.$$

In addition to the result of the entire forgotten index, using Table 5 we obtain $M{F}^{\mathcal{E}}\left(G\right)=32(p+q)+98.$ □

Table 5. The vertex edge partition incidence in the $(p,q)$-kite graph.

${\mathcal{E}}_{{\mathit{d}}_{\mathit{v}},{\mathit{d}}_{\mathit{e}}},\mathit{where}\mathit{v}\mathit{is}\mathit{incident}\mathit{to}\mathit{e}$	Number of pairs
${\mathcal{E}}_{1,1}$	1
${\mathcal{E}}_{2,1}$	1
${\mathcal{E}}_{2,2}$	$2p+2q-8$
${\mathcal{E}}_{3,3}$	3
${\mathcal{E}}_{2,3}$	3

Let $G_1,G_2,\dots G_m$ be a set of finite pairwise disjoint graphs with $v_i\in V\left(G_i\right).$ The bridge graph $B(G_1,G_2,\dots ,G_m)=B(G_1,G_2,\dots ,G_m;w_1,w_2,\dots w_d)$ of ${\left\{Gi\right\}}_{i=1}^m$ with respect to the vertices ${\left\{w_i\right\}}_{i=1}^m$ is the graph obtained from graphs $G_1,\dots ,G_m$ by connecting vertices $w_i$ and $w_{i+1}$ by an edge for all $i=1,2,\dots ,m-1$; see Figure 2.

Figure 2. The bridge graph over path $P_n$.

Theorem 4.

Let G be a bridge graph over path $P_n$, $n\ge 3$. Then,

i 

$F^{\mathcal{E}}\left(G\right)=2\left(8nm+40m-107\right)$,

ii 

$M{F}^{\mathcal{E}}\left(G\right)=4\left(104nm-258m+109\right).$

Proof. 

Let G be a bridge graph over path $P_n$ with $n\ge 3$, as in Figure 2. There are m vertices of degree one, $(m-2)$ vertices of degree three, and $2n-2+(m-2)(n-2)$ 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 $2n-4+(n-3)(m-2)$ of degree two. Therefore,

$$F^{\mathcal{E}}\left(G\right)=2\left(8nm+40m-107\right).$$

By applying the partition in Table 6 and the value of entire forgotten index, we obtain

$$M{F}^{\mathcal{E}}\left(G\right)=4\left(104nm-258m+109\right).$$

□

Table 6. The vertex edge partition in the bridge graph over path $P_n$.

${\mathcal{E}}_{{\mathit{d}}_{\mathit{v}},{\mathit{d}}_{\mathit{e}}},\mathit{where}\mathit{v}\mathit{is}\mathit{incident}\mathit{to}\mathit{e}$	Number of pairs
${\mathcal{E}}_{2,2}$	$2mn-6m+4$
${\mathcal{E}}_{2,1}$	m
${\mathcal{E}}_{1,1}$	m
${\mathcal{E}}_{2,3}$	m
${\mathcal{E}}_{3,3}$	m
${\mathcal{E}}_{3,4}$	$2m-6$

Theorem 5.

Let G be the m-bridge graph on cycle $C_n$, $n,m\ge 3$. Then,

i 

$F^{\mathcal{E}}\left(G\right)=384m+16mn-620$,

ii 

$M{F}^{\mathcal{E}}\left(G\right)=560m+32mn-866.$

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, $(2m-4)$ edges of degree four, and $(mn-2m)$ edges of degree two. Therefore,

$$F^{\mathcal{E}}\left(G\right)=384m+16mn-620.$$

The vertex edge partition can be written as shown in Table 7.

Table 7. The vertex edge partition in the m-bridge over a cycle $C_n$.

${\mathcal{E}}_{{\mathit{d}}_{\mathit{v}},{\mathit{d}}_{\mathit{e}}},\mathit{where}\mathit{v}\mathit{is}\mathit{incident}\mathit{to}\mathit{e}$	Number of pairs
${\mathcal{E}}_{3,5}$	2
${\mathcal{E}}_{3,3}$	4
${\mathcal{E}}_{4,5}$	2
${\mathcal{E}}_{2,4}$	$2m-4$
${\mathcal{E}}_{4,4}$	$2m-4$
${\mathcal{E}}_{4,6}$	$2m-6$
${\mathcal{E}}_{2,3}$	4
${\mathcal{E}}_{2,2}$	$2mn-4m$

By applying the partition in Table 7 and the value of entire forgotten index we obtain

$$M{F}^{\mathcal{E}}\left(G\right)=560m+32mn-866.$$

□

Theorem 6.

Let G be the m-bridge graph on a complete graph $K_n$, where $n,m\ge 3$. Then,

i 

$F^{\mathcal{E}}\left(G\right)=4n^{5}m-27n^{4}m+113n^{3}m-56n^3-170n^{2}m+60n^2+217nm-244n-46m$$+12$,

ii 

$M{F}^{\mathcal{E}}\left(G\right)=4n^{5}m-22n^{4}m+90n^{3}m-56n^3-105n^{2}m+30n^2+160nm-230n-20m-6.$

Proof. 

Let G be the m-bridge over a complete graph $K_n$. The graph G then has $nm$ vertices and ${\displaystyle \frac{m{n}^2-mn+2m-2}{2}}$ 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 $2n$, $m-3$ edges of degree $2n+2$, $2n-2$ edges of degree $2n-3$, $(n-1)(m-2)$ edges of degree $2n-1$, and $m(n-1)(n-2)/2$ edges of degree 2n-4$2n-4$. Therefore,

$$F^{\mathcal{E}}\left(G\right)=4n^{5}m-27n^{4}m+113n^{3}m-56n^3-170n^{2}m+60n^2+217nm-244n-46m+12.$$

The vertex edge partition can be written as in Table 8.

Table 8. The vertex edge partition incidence in the m-bridge over a complete graph $K_n$.

${\mathcal{E}}_{{\mathit{d}}_{\mathit{v}},{\mathit{d}}_{\mathit{e}}},\mathit{where}\mathit{v}\mathit{is}\mathit{incident}\mathit{to}\mathit{e}$	Number of pairs
${\mathcal{E}}_{n,2n-1}$	2
${\mathcal{E}}_{n,2n-3}$	$2n-2$
${\mathcal{E}}_{n+1,2n-1}$	2
${\mathcal{E}}_{n+1,2n}$	$2m-6$
${\mathcal{E}}_{n+1,2n-2}$	$(m-2)(n-1)$
${\mathcal{E}}_{n-1,2n-2}$	$(m-2)(n-1)$
${\mathcal{E}}_{n-1,2n-3}$	$2n-2$
${\mathcal{E}}_{n-1,2n-4}$	$m(n-1)(n-2)$

By applying the partition in Table 8 and the value of entire forgotten index, we obtain

$$\begin{array}{l}M{F}^{\mathcal{E}}\left(G\right)\&=4n^{5}m-27n^{4}m+113n^{3}m-56n^3-170n^{2}m+60n^2+217nm-244n-46m+12+2\left(n^2+{\left(2n-1\right)}^2\right)+\\ \left(n^2+{\left(2n-3\right)}^2\right)\left(2n-2\right)+2\left({\left(n+1\right)}^2+{\left(2n-1\right)}^2\right)+\left({\left(n+1\right)}^2+4n^2\right)\left(2m-6\right)+\left({\left(n+1\right)}^2+{(2n-2)}^2\right)(m-2)(n-\\ 1)+\left({\left(n-1\right)}^2+{(2n-2)}^2\right)(m-2)(n-1)+\left({\left(n-1\right)}^2+{(2n-3)}^2\right)(2n-2)+m\left({\left(n-1\right)}^2+{(2n-4)}^2\right)(n-1)(n-2)\\ =4n^{5}m-22n^{4}m+90n^{3}m-56n^3-105n^{2}m+30n^2+160nm-230n-20m-6.\end{array}$$

□

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 $S\left(G\right)$.

Theorem 7.

Let G be any graph of n vertices and m edges, and let H be the subdivision of G. Then,

i 

$F^{\mathcal{E}}\left(H\right)=F\left(G\right)+Y\left(G\right)+8m$,

ii 

$M{F}^{\mathcal{E}}\left(H\right)=4F\left(G\right)+Y\left(G\right)+16m$.

Proof. 

Let G be any graph of m edges and let H be the subdivision of G. Then,

$$\begin{array}{c}{F}^{ϵ}\left(H\right)=\sum _{x\in V\left(H\right)\cup E\left(H\right)}{d}_x^3=\sum _{v\in V\left(H\right)}{d}_v^3+\sum _{e\in E\left(H\right)}{d}_e^3\\ =\sum _{uv\in E\left(H\right)}(d_u^2+d_v^2)+\sum _{ef\in E\left(L\right(H\left)\right)}(d_e^2+d_f^2).\end{array}$$

Through a rigorous analysis of the subdivision graph definition and subsequent calculations, we determined that

$$\begin{array}{c}=\sum _{v\in V\left(G\right)}{d}_{v/G}\left(d_{v/G}^2+4\right)+\sum _{v\in V\left(G\right)}\left(d_{v/G}^2-d_{v/G}\right)d_{v/G}^2+\sum _{uv\in E\left(G\right)}(d_{u/G}^2+d_{v/G}^2)\\ =F\left(G\right)+Y\left(G\right)+8m.\end{array}$$

To determine the formula of the modified entire forgotten index, we have

$$\begin{array}{c}M{F}^{\mathcal{E}}\left(H\right)=F^{\mathcal{E}}\left(H\right)+\sum _{wincidente}(d_{w/H}^2+d_{e/H}^2)\\ =F^{\mathcal{E}}\left(H\right)+\sum _{v\in V\left(G\right)}{d}_{v/G}\left(2d_{v/G}^2\right)+\sum _{vu\in E\left(G\right)}(d_{v/G}^2+4+d_{u/G}^2+4)\\ =F^{\mathcal{E}}\left(H\right)+3F\left(G\right)+8m.\end{array}$$

Hence, $M{F}^{\mathcal{E}}\left(H\right)=4F\left(G\right)+Y\left(G\right)+16m.$ □

Theorem 8.

Let G be a graph of n vertices and m edges, and let H be its central graph. Then,

i 

$F^{\mathcal{E}}\left(H\right)=70m-27n^4+69n^3-77n^2+31n-6m{n}^3+42m{n}^2-90mn+4n^5,$

ii 

$M{F}^{\mathcal{E}}\left(H\right)=50m-22n^4+46n^3-42n^2+14n-6m{n}^3+38m{n}^2-66mn+4n^5.$

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 $2m$ edges of degree $n-1$ and $\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\sum _{v\in V\left(G\right)}(n-d_{v/G}-1)\end{array}$ edges of degree $(2n-4)$. Thus,

$$\begin{array}{c}{F}^{\mathcal{E}}\left(H\right)=8m+n{(n-1)}^3+2m{(n-1)}^3+{\displaystyle \frac{1}{2}}{(2n-4)}^3\sum _{v\in V_{(}G)}(n-d_{v/G}-1)\\ =8m+n{(n-1)}^3+2m{(n-1)}^3+4{(n-2)}^3(n^2-2m-n)\\ =70m-27n^4+69n^3-77n^2+31n-6m{n}^3+42m{n}^2-90mn+4n^5.\end{array}$$

To obtain the expression of the modified entire forgotten index, we have

$$\begin{array}{c}M{F}^{\mathcal{E}}\left(H\right)=F^{\mathcal{E}}\left(H\right)+\sum _{wincidente}(d_{w/H}^2+d_{e/H}^2)\\ =F^{\mathcal{E}}\left(H\right)+(8+2{(n-1)}^2)m+2{(n-1)}^2\sum _{v\in V\left(G\right)}{d}_{v/G}\\ +({(n-1)}^2+{(2n-4)}^2)\sum _{v\in V\left(G\right)}(n-d(v/G)-1)\\ =F^{\mathcal{E}}\left(H\right)+(8+2{(n-1)}^2)m+4m{(n-1)}^2+(n^2-2m-n)({(n-1)}^2+{(2n-4)}^2)\\ =F^{\mathcal{E}}\left(H\right)+5n^4-23n^3+35n^2-17n-4n^{2}m+24nm-20m.\end{array}$$

Hence,

$$M{F}^{\mathcal{E}}\left(B_m\right)=50m-22n^4+46n^3-42n^2+14n-6m{n}^3+38m{n}^2-66mn+4n^5.$$

□

Theorem 9.

For any book graph $B_m$, where $m\ge 3$,

i 

$F^{\mathcal{E}}\left(B_m\right)=2m^4+8m^3-24m^2+48m-32.$

ii 

$M{F}^{\mathcal{E}}\left(B_m\right)=2m^4+16m^3-22m^2+56m-48.$

Proof. 

Obviously, from the definition of the book graph $B_m$, there are $2m-2$ vertices with degree 2, two vertices of degree m, $(2m-2)$ edges of degree m, one edge of degree $(2m-2)$, and m edges of degree $(2m-2)$. Therefore,

$$F^{\mathcal{E}}\left(B_m\right)=2m^4+8m^3-24m^2+48m-32.$$

Furthermore, employing Table 9, it can be observed that

$$M{F}^{\mathcal{E}}\left(B_m\right)=2m^4+16m^3-22m^2+56m-48.$$

□

Table 9. The partition of the vertices incident with the edges in the book graph.

${\mathcal{E}}_{{\mathit{d}}_{\mathit{v}},{\mathit{d}}_{\mathit{e}}},\mathit{where}\mathit{v}\mathit{is}\mathit{incident}\mathit{to}\mathit{e}$	Number of pairs
${\mathcal{E}}_{2,m}$	$2m-2$
${\mathcal{E}}_{2,2}$	$2m-2$
${\mathcal{E}}_{m,m}$	$2m-2$
${\mathcal{E}}_{m,2m-2}$	2

Theorem 10.

Let $H\cong B_{a,b}$ with $a,b\ge 3$ be a stacked book graph. Then,

i 

$F^{\mathcal{E}}\left(H\right)=28a^3+a^{4}b-114a^2-126a+90ab-98b+14a^{3}b+9a^{2}b+182,$

ii 

$M{F}^{\mathcal{E}}\left(H\right)=28a^3+a^{4}b-144a^2-246a+152ab-164b+17a^{3}b+26a^{2}b+298.$

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 $a+1$, $2a-2$ vertices of degree 2, and $(ab-2a-b+2)$ vertices of degree 3. Regarding the edges, there are $2a-2$ edges of degree a, $(ab-2a-b+2)$ edges of degree $a+2$, $2a-2)$ edges of degree 3, $(ab-3a-b+3)$ edges of degree 4, two edges of degree $2a-1$, and $(b-3)$ edges of degree $2a$. Consequently,

$$\begin{array}{c}\hfill F^{\mathcal{E}}\left(H\right)=2a^3+{\left(a+1\right)}^3\left(b-2\right)+8\left(2a-2\right)+27\left(ab-2a-b+2\right)\\ \hfill +a^3\left(2a-2\right)+{\left(a+2\right)}^3\left(ab-2a-b+2\right)+27\left(2a-2\right)\\ \hfill +64\left(ab-3a-b+3\right)+8{\left(2a-1\right)}^3\\ \hfill +8a^3\left(b-3\right).\end{array}$$

Hence,

$$\begin{array}{c}\hfill F^{\mathcal{E}}\left(H\right)=28a^3+a^{4}b-114a^2-126a+90ab-98b+14a^{3}b+9a^{2}b+182.\end{array}$$

Based on the partition of vertices incident with edges delineated in Table 10, we deduce that

$$\begin{array}{c}\hfill M{F}^{\mathcal{E}}\left(H\right)=F^{\mathcal{E}}\left(H\right)+3\left(2a-2\right)+\left(4+a^2\right)\left(2a-2\right)+\left(4a^2+{\left(a+1\right)}^2\right)\left(2b-6\right)+18\left(2a-2\right)\\ \hfill +\left(9+{\left(a+2\right)}^2\right)\left(ab-2a-b+2\right)+25\left(2ab-2b-6a+6\right)+2\left(a^2+{\left(2a-1\right)}^2\right)\\ \hfill +\left({\left(a+1\right)}^2+{\left(a+2\right)}^2\right)\left(ba-b-2a+2\right)\\ \hfill +2\left({\left(a+1\right)}^2+{\left(2a-1\right)}^2\right)\\ \hfill +2a^2\left(2a-2\right).\end{array}$$

Hence, after some calculations, we have

$$M{F}^{\mathcal{E}}\left(H\right)=28a^3+a^{4}b-144a^2-246a+152ab-164b+17a^{3}b+26a^{2}b+298.$$

□

Table 10. The partition of the vertices incident with the edges in the stacked book graph.

${\mathcal{E}}_{{\mathit{d}}_{\mathit{v}},{\mathit{d}}_{\mathit{e}}},\mathit{where}\mathit{v}\mathit{is}\mathit{incident}\mathit{to}\mathit{e}$	Number of pairs
${\mathcal{E}}_{2,3}$	$2a-2$
${\mathcal{E}}_{2,a}$	$2a-2$
${\mathcal{E}}_{a+1,2a}$	$2b-6$
${\mathcal{E}}_{3,3}$	$2a-2$
${\mathcal{E}}_{3,a+2}$	$ab-2a-b+2$
${\mathcal{E}}_{3,4}$	$2ba-2b-6a+6$
${\mathcal{E}}_{a,2a-1}$	2
${\mathcal{E}}_{a+1,a+2}$	$ba-b-2a+2$
${\mathcal{E}}_{a+1,2a-1}$	2
${\mathcal{E}}_{a,a}$	$2a-2$

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 $n=x+2y+2z+1$ vertices and $m=x+2y+3z$ edges, where x, y, and z are positive integers.

Theorem 11.

Consider $H\cong F_{x,y,z}$, which is a firefly graph. Then,

i 

$F^{\mathcal{E}}\left(H\right)=x{(x+y+2z-1)}^3+(y+2z+1){(x+y+2z)}^3+x+10y+24z,$

ii 

$M{F}^{\mathcal{E}}\left(H\right)=x{(x+y+2z-1)}^3+(y+2z+1){(x+y+2z)}^3+\left(2x+3y+6z\right){(x+y+2z)}^2+x{\left(x+y+2z-1\right)}^2+2x+21y+48z.$

Proof. 

The firefly graph H has the following properties: for vertices, $x+y$ vertices of degree 1, $y+2z$ vertices of degree 2, and one vertex of degree $x+y+2z$; for edges, b edges of degree 1, z edges of degree 2, x edges of degree $x+y+2z-1$, and $y+2z$ edges of degree $x+y+2z$.

Therefore,

$\begin{array}{c}\hfill F^{\mathcal{E}}\left(H\right)=x+y+8(y+2z)+{(x+y+2z)}^3+y+8z+x{(x+y+2z-1)}^3\\ \hfill +(y+2z){(x+y+2z)}^3\end{array}$

$$F^{\mathcal{E}}\left(H\right)=x{(x+y+2z-1)}^3+(y+2z+1){(x+y+2z)}^3+x+10y+24z.$$

From the analysis of the vertex partition incident with the edges outlined in Table 11, after some calculations we can conclude that

Table 11. The vertex partition incident with the edges in the firefly graph.

${\mathcal{E}}_{{\mathit{d}}_{\mathit{v}},{\mathit{d}}_{\mathit{e}}},\mathit{where}\mathit{v}\mathit{is}\mathit{incident}\mathit{to}\mathit{e}$	Number of pairs
${\mathcal{E}}_{1,x+y+2z-1}$	x
${\mathcal{E}}_{1,1}$	y
${\mathcal{E}}_{2,1}$	y
${\mathcal{E}}_{2,x+y+2z}$	$y+2z$
${\mathcal{E}}_{2,2}$	$2z$
${\mathcal{E}}_{x+y+2z,x+y+2z}$	$y+2z$
${\mathcal{E}}_{x+y+2z,x+y+2z-1}$	x

$\begin{array}{c}\hfill M{F}^{\mathcal{E}}\left(H\right)=F^{\mathcal{E}}\left(H\right)+x({(x+y+2z)}^2+1)+7y+(y+2z)(4+{(x+y+2z)}^2)+16z\\ \hfill +2(y+2z){(x+y+2z)}^2+x\\ \hfill =F^{\mathcal{E}}\left(H\right)+\left(2x+3y+6z\right){\left(x+y+2z\right)}^2+x{\left(x+y+2z-1\right)}^2+x+11y+24z\\ \hfill =x{(x+y+2z-1)}^3+(y+2z+1){(x+y+2z)}^3+x+10y+24z\\ \hfill +\left(2x+3y+6z\right){\left(x+y+2z\right)}^2+x{\left(x+y+2z-1\right)}^2+x+11y+24z.\end{array}$ □

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.

Figure 3. Hydrogen-deleted molecular graphs of octane isomers.

Table 12. Experimental physico-chemical properties (S, HVAP, AF) and theoretical indices for octane isomers.

Octanes	F	EF	MEF	S	HVAP	AF
n-Oct(1)	50	92	186	111.67	73.19	0.397898
2-M-Hept(2)	62	130	252	109.84	70.3	0.377916
3-M-Hept(3)	62	142	268	111.26	71.3	0.371002
4-M-Hept(4)	62	142	268	109.32	70.91	0.371504
3-E-Hex(5)	62	154	284	109.43	71.7	0.362472
2,2-M-Hex(6)	92	254	450	103.42	67.7	0.339426
2,3-M-Hex(7)	74	198	356	108.02	70.2	0.348247
2,4-M-Hex(8)	74	180	334	106.98	68.5	0.344223
2,5-M-Hex(9)	74	168	318	105.72	68.6	0.35683
3,3-M-Hex(10)	92	192	488	104.74	68.5	0.322596
3,4-M-Hex(11)	74	210	372	106.59	70.2	0.340345
2-M-3-E-Pent(12)	74	210	372	106.06	69.7	0.332433
3-M-3-E-Pent(13)	92	314	526	101.48	69.3	0.306899
2,2,3-M-Pent(14)	104	346	582	101.31	67.3	0.300816
2,2,4-M-Pent(15)	104	292	516	104.09	64.87	0.30537
2,3,3-M-Pent(16)	104	364	604	102.06	68.1	0.293177
2,3,4-M-Pent(17)	86	254	444	102.39	68.37	0.317422
2,2,3,3-M-But(18)	134	512	826	93.06	66.2	0.255294

Table 13. Experimental physico-chemical properties (CP, MON, DENS, and MV).

Octanes	CP	MON	DENS	MV
n-Oct(1)	24.64		0.7025	162.6050
2-M-Hept(2)	24.8	23.10	0.6980	163.6530
3-M-Hept(3)	25.6	35.00	0.7058	161.8450
4-M-Hept(4)	25.6	39.00	0.7046	162.1200
3-E-Hex(5)	25.74	52.40	0.7136	160.0760
2,2-M-Hex(6)	25.6	77.40	0.6953	164.2890
2,3-M-Hex(7)	26.6	78.90	0.7121	160.4130
2,4-M-Hex(8)	25.8	69.90	0.7004	163.0930
2,5-M-Hex(9)	25	55.70	0.6935	164.7150
3,3-M-Hex(10)	27.2	83.40	0.7100	160.8870
3,4-M-Hex(11)	27.4	81.70	0.7200	158.6530
2-M-3-E-Pent(12)	27.4	88.10	0.7193	158.8070
3-M-3-E-Pent(13)	28.9	88.70	0.7274	157.0390
2,2,3-M-Pent(14)	28.2	99.90	0.7161	159.5170
2,2,4-M-Pent(15)	25.5	100.00	0.6919	165.0960
2,3,3-M-Pent(16)	29	99.40	0.7262	157.2980
2,3,4-M-Pent(17)	27.6	95.90	0.7191	158.8510
2,2,3,3-M-But(18)	24.5		0.8242	138.5980

Table 14. Correlation coefficient of $F\left(G\right)$, $F^{\mathcal{E}}\left(G\right)$, and $M{F}^{\mathcal{E}}\left(G\right)$ with entropy (S), enthalpy of vaporization (HVAP), and acentric factor (AF).

	Entropy	Enthalpy of Vaporization	Acentric Factor
$F\left(G\right)$	−0.952722167	−0.871572202	−0.96504682
$F^{\mathcal{E}}\left(G\right)$	−0.954461897	−0.749627899	−0.95719767
$M{F}^{\mathcal{E}}\left(G\right)$	−0.961615599 2	−0.789941136	−0.977606035

Table 15. Correlation coefficient of $F\left(G\right)$, $F^{\mathcal{E}}\left(G\right)$, and $M{F}^{\mathcal{E}}\left(G\right)$ with heat capacity at constant pressure (CP), motor octane number (MON), density (DENS), and molar volume (MV).

	Heat Capacity	Motor Octane Number	Density	Molar Volume
$F\left(G\right)$	0.274159212	0.850420875	0.648498071	−0.642118529
$F^{\mathcal{E}}\left(G\right)$	0.32319087	0.843802577	0.75780613	−0.75794845
$M{F}^{\mathcal{E}}\left(G\right)$	0.345514406	0.871862227	0.725891842	−0.725790445

A comparison of the correlations between the modified entire forgotten index $M{F}^{\mathcal{E}}\left(G\right)$, the forgotten index $F\left(G\right)$, and the entire forgotten index $F^{\mathcal{E}}\left(G\right)$ 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 $M{F}^{\mathcal{E}}\left(G\right)$ 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.

Figure 4. Linear relation of modified entire forgotten index with different properties of octanes.

This finding underscores the superior predictive power of our newly introduced index in characterizing molecular properties.

We obtained the following regression equations:

$$\begin{array}{c}\hfill S=-0.027476386\ast MEF\left(G\right)+116.7793982,\\ \hfill HVAP=-0.010123825\ast MEF\left(G\right)+73.35122234,\\ \hfill AF=-0.000219195\ast MEF\left(G\right)+0.426444322,\\ \hfill CP=0.00314114\ast MEF\left(G\right)+25.09394843,\\ \hfill MON=25.0323908\ast MEF\left(G\right)-2.021300133,\\ \hfill DENS=0.000134227\ast MEF\left(G\right)+0.660030483,\\ \hfill MV=-0.026899573\ast MEF\left(G\right)+170.9916232.\end{array}$$

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), $\pi$-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.

Figure 5. Molecular structures of 21 lower benzenoid hydrocarbons.

Table 16. Forgotten, entire forgotten, and modified entire forgotten topological indices along with experimental boiling point (BP), $\pi -$$\pi$-electron energy, molecular weight (MW), polarizability (P), and molar refractivity (MR) for 21 lower benzenoid hydrocarbons.

Compounds	F	EF	MEF	BP	BI-ELE	MW	PO	MR
Naphthalene	118	338	608	218	13.683	128.17	17.5	44.1
Phenanthrene	188	598	1046	338	19.448	178.23	24.6	61.9
Anthracene	188	580	1024	340	19.314	178.23	24.6	61.9
Chrysene	258	858	1484	431	25.192	228.3	31.6	79.8
Benzo[a]anthracene	258	840	1462	425	25.101	228.3	31.6	79.8
Triphenylene	258	876	1506	429	25.275	228.3	31.6	79.8
Tetracene	258	822	1440	440	25.188	228.3	31.6	79.8
Benzo[a]pyrene	312	1086	1858	496	28.222	252.3	35.8	90.3
Benzo[e]pyrene	312	1104	1880	493	28.336	252.3	35.8	90.3
Perylene	312	1104	1880	497	28.245	252.3	35.8	90.3
Anthanthrene	366	1314	2232	547	31.253	276.3	40	100.8
Benzo[ghi]perylene	366	1332	2254	542	31.425	276.3	40	100.8
Dibenz[a,c]anthracene	328	1118	1922	535	30.942	278.3	38.7	97.6
Dibenz[a,h]anthracene	328	1100	1900	535	30.881	278.3	38.7	97.6
Dibenz[a,j]anthracene	328	1100	1900	531	30.88	278.3	38.7	97.6
Picene	328	1118	1922	519	30.943	278.3	38.7	97.6
Coronene	420	1560	2628	590	34.572	300.4	44.1	111.4
Dibenzo[a,h]pyrene	382	1346	2296	596	33.928	302.4	42.9	108.1
Dibenzo[a,i]pyrene	382	1346	2296	594	33.954	302.4	42.9	108.1
Dibenzo[a,l]pyrene	382	1364	2318	595	34.031	302.4	42.9	108.1
Pyrene	242	807	1401	393	22.506	202.25	28.7	72.5

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.

Table 17. Correlation analysis of forgotten, entire forgotten, and modified entire forgotten topological indices with physicochemical properties of benzenoid hydrocarbons.

	BP	$\mathit{\pi }$-Electron Energy	MW	PO	MR
$F\left(G\right)$	0.986625	0.987801	0.979382	0.992853	0.992853
$F^{\mathcal{E}}\left(G\right)$	0.970823	0.972134	0.959881	0.979636	0.979682
$M{F}^{\mathcal{E}}\left(G\right)$	0.974955	0.976175	0.964739	0.983164	0.983215

Figure 6. Predictive power of the modified entire forgotten index, showing correlations with 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), $\pi$-electron energy, molecular weight (MW), polarizability (P), and molar refractivity (MR) of 21 lower benzenoid hydrocarbons, as follows:

$\begin{array}{c}\hfill BP=0.189807331602268\left[MEF\right]+144.14744242145,\\ \hfill \pi electronenergy=0.0779174749698\left[MEF\right]+7.69640713153,\\ \hfill MW=0.10240360571495\left[MEF\right]+110.267097355341,\\ \hfill PO=0.014925077683291\left[MEF\right]+13.02714794123,\\ \hfill MR=0.037638697447096\left[MEF\right]+32.841456251953.\end{array}$

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.