The Edge-Weighted Graph Entropy Using Redefined Zagreb Indices

Measurements of graphs and retrieving structural information of complex networks using degree-based network entropy have become an informational theoretical concept. This terminology is extended by the concept of Shannon entropy. In this paper, we introduce entropy with graphs having edge weights which are basically redeﬁned Zagreb indices. Some bounds are calculated to idealize the performance in limiting diﬀerent kinds of graph entropy. In addition, we discuss the structural complexity of connected graphs representing chemical structures. In this article, we have discussed the edge-weighted graph entropy with ﬁxed number of vertices, with minimum and maximum degree of a vertex, regular graphs, complete graphs, complete bipartite graphs, and graphs associated with chemical structures.


Introduction
In network theory, entropy measure has become an interesting field during last 5 decades due to Shannon [1]. Rashevsky [2] defined partitions of vertex orbits to study structural information. Later on, Mowshowitz [3] used the same concept to prove certain properties for some operations of graphs (see [4]). Graph entropy measurement was used by Rashevsky to study structural complexity of graphs based on Shannon's entropy. Mowshowitz [3] introduced applications of graph entropy in information technology. e other applicaitons of graph entropy include characterizing graph patterns in chemistry, biology, and computer scineces [5]. In the literature, the graph entropy was defined in different ways. An important example is Körner's entropy [6] introduced with an information theory-specific point of view.
e basic properties of graph entropy were discussed in [7]. e graph entropy satisfies subadditive property in union of graphs which can be extended in graph covering problem and the problem of perfect hashing.
Graph entropy used for minimum number of perfect hash functions over a given range of hash for all k-subsets of a set of given size is discussed in [8]. Another application of graph entropy is due to Kahn and Kim [9], who proposed an algorithm based on graph entropy of an appropriate comparability graph. Csiszar et al. [10] in 1990 characterized minimal pairs of convex corners which generate the probability density p � (p 1 , p 2 , . . . , p k ) ∈ in a k− dimensional space. Due to this fact, another definition of graph entropy in terms of vertex packing polytope of a graph was introduced. ey also gave characterization of a perfect graph using the subadditivity property of graph entropy (see [6]). eir examinations prompted the thought of a class of graphs called normal graphs, a generalization of perfect graphs given in [11]. Alon and Orlitsky [12] examined the connection between the base entropy shading of a graph and the graph entropy. Distinctive graph invariants are utilized to create graph entropy estimates, for example, eigenvalue and network data [13], weighted graph entropy with distance-based TIs as edge weights [14], and weighted graph entropy with degree-based TIs as edge weights [15]. ere are various uses of graph entropy in interchanges and financial matters.
Different graph invariants such as connectivity information, distance-based topological indices (Wiener related index), degree-based topological indices, the no. of vertices, the vertex degree sequences, the second neighbor degree sequences, the third neighbor degree sequences, eigenvalues, and so on were used to define graph entropy in the literature (see [16]). e properties of graph entropies based on information functionals by using degree powers of graphs have been explored in [17,18]. e most important graph invariants are the Zagreb index [19] and the zeroth-order Randic index [20]. Chen et al. [21] proposed the idea of graph entropy for unique weighted graphs by utilizing Randić edge weights and demonstrated extremal properties of graph entropy for some basic groups. Kwun et al. [22] established the weighted entropy with atomic bound connectivity edge weights.
In this paper, the idea presented in [23] is used to study the weighted graph entropy by using redefined Zagreb indices as edge weights. Some degree-based indices are portrayed by researching the boundaries of the entropy of certain class of graphs [22]. Our aim is also to address the issue proposed by Chen et al. in [21] and Kwun et al. in [24]. e paper is organized into different sections to discuss the proposed problems.

Topological Index
In chemical structures, various graph notions used are molecular descriptors (or topological index). Few of them are first and second Zagreb indices M 1 (G) and M 2 (G), respectively, exhibiting applicability for deriving multi-linear regression models. Details on the chemical applications of these indices can be seen in [25,26]. More results on Zagreb indices can be seen in [27,28]. Applications of Zagreb indices in QSPR were exposed by modeling the structure-boiling point associated with C 3 − C 8 alkanes using the CROMRsel method [29,30]. A large number of studies have been conducted on these two indices.
Ranjini et al. [31] introduced redefined first, second, and third Zagreb indices as

Entropy
e entropy of a graph is a functional depending both on the graph itself and on a probability distribution on its vertex set. is graph functional originated from the problem of source coding in information theory and was introduced by Körner in 1973.
Consider a graph Γ having order q with degree of a vertex x denoted by d x . For any edge xy, the probability density function is defined as where w(xy) is the weight xy and w(xy > 0). e weighted graph entropy is defined as Extending this method, we introduced entropy of the edge-weighted graph. For an edge-weighted graph, Γ � (x, E, w), where x, E, and w denote vertex sets, edge sets, and edge weights of Γ, respectively, where the edge weight is positive.

Main Results
In this section, we extend the idea of graph entropy as edgeweighted graphs explained in [23] that tackled the issue of weighted synthetic graph entropy. We also addressed the issue proposed by Chen et al. in [21] and Kwun et al. in [24]. Now we present bounds of the weighted entropy for the connected graphs, regular graphs, complete graphs, complete bipartite graph,s and graphs associated with chemical structures. In these results, the edge weights are redefined first, second, and third Zagreb indices.

Theorem 1.
Let Γ be a connected graph with q, q ≥ 3 vertices; then, we have

Proof
(1) In simple connected graph Γ, of order q, the maximum and minimum degrees for a vertex are q − 1 and 1, respectively. erefore, for any edge xy, the minimum and maximum possible degrees of x and y are 1, 2 and q − 1, q − 1, respectively. erefore, we have Also, Hence, Mathematical Problems in Engineering 3 e proofs of (2) and (3) follow the proof of (1). □ Theorem 2. Let Γ be a graph having q vertices with minimum and maximum degrees of c being δ and Δ, respectively. en, we have

Proof
(1) For a connected graph of order q, the minimum and maximum degrees for a vertex are 1 and q − 1, respectively. For any edge xy, the minimum and maximum possible degrees of vertices x and y are 1, 2 and q − 1, q − 1, respectively. erefore, we have Also, Hence, e proofs of (2) and (3) follow the proof of (1).
Proof. For a t regular graph t ≥ 2, as Γ is connected with q ≥ 3,

Mathematical Problems in Engineering
Also � log RZ 1 . Hence, e proofs of (2) and (3) follow the proof of (1).

□ Theorem 4. Let Γ � (V, E, w) be an arbitrary complete graph of order q; then, we have
Proof. For an arbitrary complete graph Γ of order q, we have RZ 1 ≤ (q 2 /2); therefore, the result is obvious.

Proof
(1) Assume that Γ is a complete bipartite graph with q vertices with two parts having s and t vertices, respectively. erefore, we have s + t � q, and we have For minimum value, we take s � 1 and a � q − 1, while for maximum value, s � a � (q/2), and hence e proofs of (2) and (3) follow the proof of (1).

□
Chemical graphs are associated with structure of chemical compounds. erefore, we consider atoms as vertices and chemical bonds as edges of that graph. e following theorem provides bounds for weighted graph entropy of chemical graph by assigning RZ 1 as edge weights.

Proof
(1) For a chemical graph Γ ′ , for any edge xy, the maximum degrees of x and y are 4 and 4 and minimum possible degrees are 1 and 2. So, we have Similarly, erefore, e proofs of (2) and (3) follow the proof of (1).

Numerical Examples
Example 1. e molecular graphs of carbon nanotubes VC 5 C 7 [p, q] are depicted in Figure 1.
e structure of VC_5C_7 [p,q] nanotubes consists of C_5 loops and C_7 net following the trivalent decoration. It can cover a cylinder or a ring. Two-dimensional lattice of VC 5 C 7 [p, q] is depicted in Figure 1. Now, we focus on calculating the entropy of a given structure. It can be observed from Figure 1 that the edge set of VC 5 C 7 [p, q] has partition given in Table 1.
erefore, the entropy of Γ is computed as 6 Mathematical Problems in Engineering Also, Moreover, Example 2. e molecular graphs of carbon nanotubes VC 5 C 7 [p, q] and HC 5 C 7 [p, q] are shown in Figures 2 and 3, respectively. e structure of these nanotubes consistes of C_5 loops and C_7 net following the trivalent decoration. It can cover a cylinder or a ring. Two-dimensional lattice of VC 5 C 7 [p, q] is depicted in Figure 2 and that of HC 5 C 7 [p, q] is depicted in Figure 3. Now, we focus on calculating the entropy of a given structure. From Figure 2, it can be observed that the edge set of VC 5 C 7 [p, q] has the following partitions.
It can be observed from Figure 3 that the edge set of HC 5 C 7 [p, q] can be divided into following three classes as given in Table 2.
Furthermore, Figure 1: Example 3. In nanosciences, SC 5 C 7 [p, q] (where p and q represent number of heptagons and number of cycles in each row separately in the entire lattice) nanotubes are a class C 5 C 7 , and this is generated by alternating C5 and C7. e standard tiling of C 5 and C 7 can cover cylinders or rings, and each covers SC 5 C 7 [p, q]. Nanotubes NPHX[p, q] (where p and q are expressed as the number of hexagon pairs in the first row and the other hexagons in the column) are trivalent modifications in the order of C 6 , C 6 , C 4 , C 6 , C 6 , C 4 , . . . e first row of the sequence is C 6 , C 8 , C 6 , C 8 , . . . in the other rows. In other words, the nanolattice can be viewed as a tile of C 4 , C 6 , and C 8 . erefore, such tiles can cover cylinders or rings.

Conclusions
In study of information theory, the graph entropy measures the information rate achievable by communicating symbols over a channel in which certain pairs of values may be confused [32][33][34][35][36]. is measure introduced by Körner in the 1970s has proven itself useful in other settings, including combinatorics.
In the present paper, we studied weighted graph entropy by using first, second, and third redefined Zagreb indices as edge weights. We also presented numerical examples to justify our results. Interesting work would be to study weighted graph entropy with other degree and distance-based topological indices.
e bounds achieved for degree-based network weighted graph entropy can be used for national security, social networks, Internet networks, structural chemistry, computational systems biology, and so on. ese studies will play an important role in asymmetry in real networks and analyzing structural symmetry in future.

Data Availability
e data used to support the findings of this study are included within this article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
Jing Lu wrote the final version of the paper and used the software to prepare graphs and verify computational results. Hafiz Mutee-ur-Rehman computed the results. Saima Nazeer wrote the first version of the paper. Xuemei An analyzed and verified the results and arranged the funding for this study.