Abstract
The graph entropy was proposed by Körner in the year 1973 when he was studying the problem of coding in information theory. The foundation of graph entropy is in information theory, but it was demonstrated to be firmly identified with some established and often examined graph-theoretic ideas. For instance, it gives an equal definition to a graph to be flawless, and it can likewise be connected to acquire lower bounds in graph covering problems. The objective of this study is to solve the open problem suggested by Kwun et al. in 2018. In this paper, we study the weighted graph entropy by taking augmented Zagreb edge weight and give bounds of it for regular, connected, bipartite, chemical, unicyclic, etc., graphs. Moreover, we compute the weighted graph entropy of certain nanotubes and plot our results to see dependence of weighted entropy on involved parameters.
1. Introduction
Due to Shannon and Weaver’s study [1], the entropy measure for investigating systems based on networks becomes an emerging field for the last 50 years. In [2], for the first time, Rashevsky introduced the structural information content that depends on the partitions of vertex orbits. Later on, the same concept was used by Mowshowitz [3], and he proved certain properties for some operations of graphs [4–6]. Rashevsky also used the graph entropy for measurement of the structural complexity of the graphs based on Shannon’s entropy. Mowshowitz, in [3], investigated the entropy of graphs as an information-theoretic quantity as the structural information content of a graph and gave some applications. Graph entropy got applications in different fields such as characterizing graph patterns in chemistry, biology, and computer science [7, 8]. In the literature, the graph entropy was defined in different ways, and an important example is Körner’s entropy introduced in [9] from an information theory-specific point of view.
The basic properties of graph entropy are discussed in [10]. The graph entropy is subadditive with respect to the union of graphs which leads the application of graph entropy in the graph covering problem and the problem of perfect hashing.
In [11], graph entropy was used for the minimum number of perfect hash functions of a given range of hash all element subsets of a set of given size. Another application of graph entropy is due to Kahn and Kim. They, in [12], proposed an algorithm that was based on graph entropy of an appropriate comparability graph. In the year 1990, Csiszar et al. [13] characterized minimal pairs of convex corners which generate the probability density in a dimensional space. Due to this fact, another definition of the entropy of the graph in terms of the vertex packing polytope of the graph was introduced. They also gave characterization of a perfect graph using the subadditivity property of graph entropy, which was further investigated in [14–17]. Their examinations prompted the thought of a class of graphs which is called normal graphs. Normal graphs are the generalization of perfect graphs [18, 19]. Alon and Orlitsky, in [20], 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 [21], weighted graph entropy with distance-based TIs as edge weights [22], and weighted graph entropy with degree-based TIs as edge weights [23]. There are various uses of graph entropy in interchanges and financial matters. We utilize the idea of graph entropy as a weighted graph similarly to [24], where the authors tackled the issue of weighted synthetic graph entropy by utilizing uncommon data utilitarian. Some degree-based indices are portrayed by researching the boundaries of the entropy of certain class of graphs [25].
In the literature, different graph invariants were used to define graph entropy, for example, 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, and eigenvalues [26–29]. The properties of graph entropies that are based on information functional by using degree powers of graphs have been explored too; see [30, 31]. The degree power is one of the most important graph invariants and well studied in graph theory; it is also related to the Zagreb index [32] and the zeroth-order Randić index [33].
In [34], Chen et al. presented 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. In [25], Kwun et al. established the weighted entropy with atomic bound connectivity edge weights. Our point is to take care of the issue proposed by Chen et al. in [34] and Kwun et al. in [35].
The paper is organized as follows: in Section 2, we give brief introduction about topological indices; in Section 3, we introduce weighted graph entropy with augmented Zagreb edge weights. In Section 4, we present some external properties of weighted graph entropy. Section 5 contains numerical examples, and in Section 6, we conclude the paper.
2. Importance of the Topological Index
Mathematical chemistry provides tools such as polynomials and functions that depend upon the information hidden in the symmetry of graphs of chemical compounds and help to predict properties of understudy molecular compounds without use of quantum mechanics [49–52]. A topological index is a numerical parameter of a graph and depicts its topology [36–38]. Degree-based topological indices correlate the structure of the molecular compound with its various physical properties, biological activities, and chemical reactivity [39]. Recently, different topological indices have been used to establish weighted graph entropies [34, 35]. Research indicates that the augmented Zagreb index (AZI) possesses the best correlating ability among several topological indices [40]. The augmented Zagreb index is defined as
In this paper, we study weighted graph entropy by taking augmented Zagreb edge weight.
3. Definition of Entropy
For a graph , the degree of vertex is denoted by and is the number of vertices attached with . For any edge , we havewhere is the edge weight assigned to the edge , , and represents the total weights of edges incident to vertex . The node entropy has been defined by
Definition 1. Consider an edge weighted graph , where , and represent the vertex set of , edge set of , and edge weights, respectively; then, the weighted entropy is defined bywhere .
In this article, we suppose that the edge weights are positive, and we take .
4. Bounds of Weighted Graph Entropy with Edge Weights
In this section, we proved some bounds for the entropy for special kinds of graphs, for example, connected graphs, regular graphs, complete bipartite graphs, chemical graphs, and unicyclic graphs. In the following theorem, we present bounds of weighted entropy for simple connected graphs.
Theorem 1. For a simple connected graph , with vertices for , we have
Proof 1. In a simple connected graph , which has vertices, the maximum and minimum degree for a vertex is and 1, respectively, so for any edge , the minimum possible degrees of and are 1 and 2, and the maximum possible degrees of and are and ; so, we haveAlso,Hence,
Theorem 2. Let us consider a graph having vertices such that and be the minimum and maximum degree of , respectively. Then, we have
Proof 2. For a connected graph of order , the maximum degree for a vertex is , and the minimum degree is 1. With any edge , the minimum possible degrees of and are 1 and 2, respectively, and the maximum possible degrees of and are and ; so, we haveAlso,Hence,
Theorem 3. For a regular graph with vertices such that , we have
if and only if is a cyclic graph, and if and only if is a complete graph.
Proof 3. Let be a regular graph, . As is connected with ,Also,
Theorem 4. For a complete bipartite graph with vertices, we have
Also, if and only if is a star graph, and if and only if is a complete bipartite graph (balanced).
Proof 4. For a complete bipartite graph with vertices and having two parts with and vertices, respectively. Therefore, we have , and we haveFor the minimum value, we take and , while for the maximum value, ; hence,The left equality holds if and only if is a star graph, and the right equality holds if and only if is a complete bipartite graph (balanced).
Chemical graph is associated with the chemical compound in which atoms are taken as vertices and chemical bonds are taken as edges. In the following theorem, we give bounds for the weighted entropy of the chemical graph by taking edge weights.
Theorem 5. Let be a chemical graph with vertices; then, we have
Proof 5. For a chemical graph , for any edge , the maximum degrees of i and are 4 and 4, and the minimum possible degrees are 1 and 2. So, we haveSimilarly,Therefore,
Corollary 1. Let us consider a chemical bicyclic graph ; then, we have
Proof 6. For a chemical graph , we haveSo, we have
Corollary 2. Let us consider a chemical bicyclic graph ; then,
Proof 7. For a chemical graph , we haveSo,
Corollary 3. Let us consider a chemical bicyclic graph ; then,
Proof 8. For a chemical graph , we haveSo,
5. Numerical Examples
Firstly, we constructed graphs associated with the concerned nanotubes and count the number of edges and vertices. Then, we divide the edge sets of concerned nanotubes into different classes with respect to the degree of end vertices. Lastly, we compute the entropies of underlined nanotubes and plot the graph of our computed results by using Maple to see the dependence of our results on the involved parameters.
5.1. Nanotubes
The term “nano” means one billionth of something. “Nano” can be credited to any unit of measure. For example, you may report somewhat mass in nanograms or the proportion of liquid in one cell to the extent nanoliters. Nanoscience is the examination of structures and materials on the measure of nanometers. At the point when structures are sufficiently made in the nanometer measure go, they can go facing captivating and important properties. Nanoscale structures have existed in nature a long time before analysts began thinking about them in labs. Analysts have even made nanostructures in the lab that duplicate a bit of nature’s shocking nanostructures [41, 42].
Nanoscience has adequately influenced our lives with improvements, for instance, recolor safe textures moved by nanoscale features found on lotus plants and PC hard drives, which store information on attractive strips that are just 20 nanometers thick. Researcher use nanotubes in figuring, prescription, vitality, data stockpiling, and so on. Carbon nanotubes can be used as the pores in layers to run invert assimilation desalination plants. Water molecules experience the smoother dividers of carbon nanotubes more easily than through various sorts of nanopores, which require less power. Various researchers are using carbon nanotubes to develop little, reasonable water-cleaning gadgets required in poor countries. Sensors using carbon nanotube recognition components are fit for recognizing an extent of concoction vapors. These sensors work by reacting to the adjustments in the obstruction of a carbon nanotube inside seeing a compound vapor [43].
The features of energetics and electronic properties of carbon nanotubes, containing a pentagon-heptagon pair (5/7) topological imperfection in the hexagonal arrangement of the crisscross setup, are investigated by using the extended Su-Schriffer–Heeger model in perspective on the tight limiting estimation in the genuine space. The estimations exhibit that this pentagon-heptagon set distortion in the nanotube structures is not in charge of a change in the nanotube measurement, yet, moreover, administers the electronic conduct around Fermi level [44].
The objective of this section is to compute weighted entropies of three nanotubes, , , and . In nanoscience, nanotubes (where and are denoted as the number of squares in a row and the number of squares in a column, resp.) are plane tiling of . This tessellation of can cover either a torus or a cylinder. The 3D representation of is described in Figure 1.
The other two understudy nanotubes are zigzag and armchair , where is the number of hexagons in the first row and is the number of hexagons in the first column. The molecular structures can be referred to Figures 2 and 3, respectively [45].
Example 1. Assume that is the graph of , as shown in Figure 1. Then, using the edge partition of given in Table 1,
From Figure 4, one can observe the behaviour of entropy of . It can be observed that the entropy of increases exponential with respect to the involved parameters and .
Example 2. Assume a graph of , as shown in Figure 2. Then, using the edge partition of given in Table 2,
From Figure 5, the behaviour of entropy of can be observed. The entropy of also increases with increment in and .
Example 3. Assume that is the graph of , as shown in Figure 3. Then, using the edge partition of given in Table 3,
From Figure 6, one can observe the behaviour of entropy of .
6. Concluding Remarks
Weighted entropy is a special case of Shannon’s entropy and is the proportion of data provided by a probabilistic analysis whose basic occasions are portrayed both by their target probabilities and by some subjective (objective or emotional) loads. It is helpful to rank synthetic substances in quantitative high-throughput screening tests [46, 47] and might be utilized to adjust the measure of data and the level of homogeneity related to a segment of information in classes [48]. Weighted entropy additionally discovered applications in the coding theory [47]. In this paper, we have studied weighted entropy with augmented Zagreb edge weights, which was an open problem [35].
Data Availability
All the data required for this research are included within this article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to this paper.
Acknowledgments
This research was funded by the Higher Education Commission, Pakistan.