Abstract
Motivated by the concept of Shannon’s entropy, the degree-dependent weighted graph entropy was defined which is now become a tool for measurement of structural information of complex graph networks. The aim of this paper is to study weighted graph entropy. We used and Gaurava indices as edge weights to define weighted graph entropy and establish some bounds for different families of graphs. Moreover, we compute the defined weighted entropies for molecular graphs of some dendrimer structures.
1. Introduction
The branch of mathematics known as graph theory provides tools for solving problems of information theory, computer sciences, physics, and chemistry [1–3]. Among them, special places are reserved for so-called topological descriptors, which play an important role in mathematical chemistry, especially in QSPR/QSAR surveys. Many topological descriptors are introduced and studied in the literature, such as the Zagreb index [4–6], the Randić connectivity index [7], and the modified Zagreb index [8, 9].
In the year 2009, Vukičević and Furtula [10] introduced the geometrical arithmetic (GA) index as a molecular descriptor, and mathematical formula for this index is,
There are many interesting attributes on topological indices, and different physicochemical properties of hydrocarbons can be obtained from this index [11–16]. The predictive power of the GA index is compared with others such as the famous Randić index [17]. Due to this reason, different versions of the GA index are now investigated and introduced in the literature [18–20]. The other famous indices are given in [21]. The mathematical formulae for first and second Gaurava indices arerespectively.
Many problems of information theory, biology, computer sciences, chemistry, and discrete mathematics are directly solved by utilizing different kinds of graph measures and the graph entropy is one of the powerful tools [22–26] that help to understand the structural complexity of graph networks [27]. Different molecular descriptors are used to introduce weighted graph entropies [28–31].
In this paper, we extended the work of [28, 30] and introduced weighted graph entropies by using GA and Gaurava indices as edge weights. We examine the extreme properties of these entropies for some special graph families. We also computed these entropies for different chemical structures.
Now, we define some basic notions about entropy of graphs. We always consider being a connected graph with E as the set of edges, as the set of vertices, and to be the edge weight given to the edges of graph that will be used to define weighted graph entropy. As mentioned above, GA and Gaurava indices are taken as edge weights in this paper. denoted the degree of a vertex u that is defined as the total number of vertices of that are at distance one from the vertex u. Considerwhere is the weight and .
Now, the weighted graph entropy can be defined by
Definition 1. For the graph , the weighted entropy can be defined as follows [28, 30]:Here, is same as that given in (4).
2. Main Results
In this section, we are going to present our main results.
Lemma 1. Let be a simple graph having vertices and edges and let and be the maximum degree and minimum degree of a vertex, respectively; then,with equality if and only if is a regular graph or is a bipartite graph.
Lemma 2. Consider is a simple connected graph having m edges. Then, we havewith equality if and only if is isomorphic to or is isomorphic to regular graph or is isomorphic to semiregular graph.
Lemma 3. Consider is a simple connected graph having m edges. Then, we haveMoreover, the equality holds if and only if is isomorphic to a regular graph or is isomorphic to semiregular graph.
Lemma 4. Consider is a simple connected graph having m edges. Then, we havewith equality if and only if is isomorphic to or is isomorphic to a complete graph .
Lemma 5. Consider is a simple connected graph having m edges. Then, we have
Theorem 1. Consider is a connected graph having n vertices with . Then, we have
Proof. We prove the result for , the other results can be proved in the same manner.
Since is a connected graph with n number of vertices, for any vertex, the maximum possible degree can be n − 1 and the minimum possible degree can be one. Hence, for any edge uv, the minimum degrees for u and can be 1 and 2 and maximum possible degrees for u and can be n − 1 and n − 1, and hence, we haveTherefore,
Theorem 2. Let be a graph with vertices. Let and be the minimum and maximum degrees of , respectively. Then, we have
Proof. Since is the connected graph with n number of vertices, for any vertex, the maximum possible degree can be n − 1 and the minimum possible degree can be one. Hence, for any edge uv, the minimum degrees for u and can be 1 and 2 and maximum possible degrees for u and can be n − 1 and n − 1, and hence, we haveAlso,
Theorem 3. Consider is a regular graph having n vertices with . Then, we haveNote that left inequality turns into equality if is a cyclic graph, and the right inequality turns into equality if is a complete graph.
Theorem 4. Consider is a complete bipartite graph having n vertices. Then, we haveNote that the left inequality turns into equality if is a star graph, and if and only if is a complete bipartite graph (balanced).
Theorem 5. Let be a tree of order with maximum degree vertex ; then, we have
Theorem 6. Let be a simple graph having vertices and edges and let and be the maximum degree and minimum degree of a vertex, respectively; then,with equality if and only if is either a regular graph or a bipartite graph.
In graph theory, the molecular graph is obtained by taking atoms as vertices and bounds as edges. It can be noted that the maximum possible degree for a vertex in a molecular graph is four. Following theorem is about the bounds of weighted entropy for the molecular graph.
Theorem 7. Consider a molecular graph having n vertices. Then, we have
2.1. Relation of Entropy with Zagreb Indices
Theorem 8. Consider is a simple connected graph. Then, we haveThe equality holds if and only if is a union of .
Theorem 9. Consider is any graph, then we haveEquality holds if and only if is isomorphic to the regular graph.
Theorem 10. Consider is any graph, then we haveEquality holds if and only if is isomorphic to the regular graph.
Theorem 11. Consider is any graph, then we haveEquality holds if and only if is isomorphic to the regular graph.
2.2. Numerical Examples
Here, we compute the weighted entropies introduced in this paper for some chemical structures.
Example 1. Consider the porphyrin dendrimers shown in Figure 1. We denote the graph of porphyrin dendrimers by , and the edge partition of is given in Table 1. Using Table 1 and definition of entropy, we have the following entropies for porphyrin dendrimers:
Example 2. The graph G of zinc-porphyrin dendrimer is shown in Figure 2, and the edge partition for this dendrimer is given in Table 2. We have the following computations for the entropies of zinc-porphyrin dendrimer.
Example 3. Let be the graph of poly(ethylene amidoamine) dendrimers as shown in Figure 3. Then, the edge partition of this dendrimer is given in Table 3 and we have the following results.
3. Conclusion
In information theory, the graph entropy is a measure of the information rate achievable by communicating symbols over a channel in which certain pairs of values may be confused. This measure, first introduced by Körner in the 1970s, has since also proven itself useful in other settings, including combinatorics. In this paper, we have studied graph entropy with and Gaurava indices and justified it by some numerical examples. It would be interesting to work on entropy of weighted graphs with some other degree- and distance-based topological indices. The bounds of degree-based network entropy can also be used in national security, Internet networks, social networks, structural chemistry, ecological networks, computational systems biology, etc. They will play an important role in analyzing structural symmetry and asymmetry in real networks in the future.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they do not have any conflicts of interest.
Authors’ Contributions
Tiejun Wu enhanced the introduction section and improved the reference list. Hafiz Mutee Ur Rehman wrote the original paper. Yu-Ming Chu gave the equality conditions for Theorems 1, 2, and 7 and verified all results. Deeba Afzal supervised the work and Jianfeng Yu prepared the revision and arranged funding for the paper.
Acknowledgments
The research was supported by the National Natural Science Foundation of China (Grant nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485). This work was also supported by the Scientific Research Project of Department of Education of Guangdong Province (natural science) (no. 2017GKTSCX102), Scientific Research Project of Department of Education of Guangdong Province (innovation) (No. 2017GGXJK095), Teaching Reform Project of Guangdong Higher Vocational Education Machinery Manufacturing Major Teaching Steering Committee (JZ201907), and Curriculum Ideological and Political Demonstration Curriculum Construction Project of Dongguan Polytechnic (KCSZ202002).