# The Entropy of Weighted Graphs with Atomic Bond Connectivity Edge Weights

Authors: Young Chel Kwun; Hafiz Mutee ur Rehman; Muhammad Yousaf; Waqas Nazeer; Shin Min Kang
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

The aim of this report to solve the open problem suggested by Chen et al. We study the graph entropy with ABC edge weights and present bounds of it for connected graphs, regular graphs, complete bipartite graphs, chemical graphs, tree, unicyclic graphs, and star graphs. Moreover, we compute the graph entropy for some families of dendrimers.


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## Body

## 1. Introduction

Mathematical chemistry is the branch of theoretical chemistry in which we discuss and predict the behavior of mathematical structure by using mathematical tools [1, 2]. In the past few decades, there have been many studies in this area. This theory has played an important role in the field of chemistry.The topological index is a real number associated with the molecular graph. It is a graph invariant. Many topological indices are defined up till now [3–5]. Some of them are based on distance, while others are based on degree and have found many applications in pharmacy, theoretical chemistry, and especially QSPR/QSAR research.In 1975, the first degree-based topological index [6] was proposed. Later, this index was generalized to any real number α by Estrada et al. in [7] and was named the generalized Randić index. Another well-known topological index based on the vertex degree of the graph is the atomic bond connectivity index [8], which is defined as (1)ABCG=∑uv∈EGdu+du-2du.duAt the beginning, a close link between the heat of alkane formation and the ABC index was experienced. After that, the ABC index became a powerful tool for simulating the thermodynamic properties of organic compounds. The fourth member of the ABC index category was proposed by M. Ghorbaniet al. in [9]. In recent years, many papers are written on topological indices and its application; here we mention few [10–15].Based on the groundbreaking work of Shannon [16], in the late 1950s began to study the entropy measurement of network systems. Rashevsky uses the concept of graph entropy to measure the structural complexity of the graph. Here, the complexity of his graph is based on Shannon’s entropy. Mowshowitz [17] introduced the entropy of the graph as information theory, which he interpreted as the structural information content of the graph. Mowshowitz [18] later studied the mathematical properties of graph entropy and conducted indepth measurements of his particular application. Graph entropy measures have been used in various disciplines, for example, to characterize patterns in biology, chemistry, and computer science. Therefore, it is not surprising to realize that “graph entropy” has been defined in various ways. Another classic example is the introduction of Körners entropy [19].Different graph invariances have been used to develop image entropy measures such as eigenvalue and connectivity information [20], distance-based graph entropy [21], and degree-based graph entropy [22].We have different applications of graph entropy in communications and economics. We use the concept of graph entropy as a weighted graph, just as Dehmer [20] who solved the problem of weighted chemical graph entropy by using a special information functional. Some degree-based indices are characterized by investigating the extremes of the entropy of certain class of graphs [23, 24].In [25], Chen et al. introduced the concept of graph entropy for special weighted graphs by using Randic edge weights and proved extremal properties of graph entropy for some elementary families of graphs. Our aim is to solve problem suggested by Chen et al. in [25]. In this paper, we study graph entropy by taking Atomic bond connectivity edge weights and prove some extermal properties of graph entropy for special families of graphs such as connected graphs, regular graphs, complete bipartite graphs, chemical graphs, tree, unicyclic graphs, and star graphs. Moreover we compute graph entropy of different dendrimer structures.


## 2. Main Results

Let us have a graph, where G andvi are its vertices and the degree of vi is denoted by di. For an edge vivj we have(2)pij=wvivj∑j=1diwvivjwhere w(vivj) is the weight vivj and w(vivj) > 0. The weighted entropy is defined as(3)Hvi=-∑j=1dipijlog⁡pij.Inspired by this method, we have introduced the definition of the entropy of the edge-weighted graph, which can also be interpreted as multiple graphs. For edge weight graphs, G = (V, E, w), where V, E, and w denote the vertex sets of G, edge sets, and edge weights (sometimes also called costs). In this article, we always assume that the edge weights are positive.Definition 1.
LetG=(V,E,w) be an edge weighted, then the entropy of G is(4)IG,w=-∑uv∈Epuvlog⁡puv.where puv=w(uv)/∑uv∈Ew(uv)Theorem 2.
For a connected graph G withn vertices for n≥3, we have (5)log⁡ABC+log⁡1n-2≤IG,ABC≤log⁡ABC-log⁡1n-1Proof.
For a simple connected graph of ordern, the maximum degree for a vertex isn-1 and minimum degree is 1. With any edge uv, the minimum possible degrees of u and v are 1 and 2, respectively, and maximum possible degrees of u and v are n-1 and n-1, so we have(6)IG,ABC=log⁡ABC-1/2ABC∑du+du-2du.dulog⁡du+du-2du.du=log⁡ABC-12ABC∑du+du-2du.dulog⁡du+dv-2-log⁡du.dv)≤log⁡ABC-12log⁡1-log⁡n-12=log⁡ABC-log⁡1n-12≤log⁡ABC-log⁡1n-1,IG,ABC≥log⁡ABC-12ABC∑du+du-2du.dulog⁡2n-4-log⁡2=log⁡ABC-12log⁡n-2=log⁡ABC+log⁡1n-2=log⁡ABC+log⁡1n-2Corollary 3.
Let G be a graph withn vertices. Let δandΔ be the minimum degree and maximum degree of G, respectively. Then we have(7)logABC⁡+log⁡δ2Δ-2≤IG,ABC≤log⁡ABC-12log⁡Δ2δ-2Proof.
(8) I G , ABC ≤ log ⁡ A B C - 1 2 A B C ∑ d u + d v - 2 d u . d v log ⁡ 2 δ - 2 - log ⁡ Δ 2 = log ⁡ A B C - 1 2 log ⁡ Δ 2 δ - 2Also(9)IG,ABC≥log⁡ABC-12log⁡2Δ-2-log⁡δ2=log⁡ABC+log⁡δ2Δ-2.Theorem 4.
For a regular graphG=(V,E,w) with n vertices such that n≥3,we have (10)log⁡n≤IG,ABC≤log⁡nn-12,and log⁡n=IG,ABC if and only if G is cycle graph, and IG,ABC=log⁡(n(n-1)/2) if and only if G is complete graph.Proof.
Let ak regular graph G with k≥2. As G is connected with n≥3, so (11)IG,ABC=-∑2k-2/k2∑2k-2/k2log⁡2k-2/k2∑2k-2/k2=-∑2nklog⁡2nk=-log⁡nk2.Since 2≤k≤n-1, we have(12)log⁡n≤IG,ABC≤log⁡nn-12Theorem 5.
For a complete bipartite graphG=(V,E,w) with n vertices, we have(13)log⁡n-1≤IG,ABC≤log⁡n2n2,and log⁡n-1=IG,ABC if and only if G is star graph, and IG,ABC=log⁡n/2n/2 if and only if G is complete bipartite graph (balanced).Proof.
For a complete bipartite graphG=(V,E,w) with n vertices and two parts have p and q vertices, respectively. Therefore p+q=n. We have(14)IG,ABC=-∑p+q-2/pq∑p+q-2/pqlogp+q-2/pq∑p+q-2/pqIG,ABC=-∑p+q-2/pqpqp+q-2/pqlog⁡p+q-2/pqpqp+q-2/pq=-∑1pqlog⁡1pq=log⁡pqThus(15)log⁡n-1≤IG,ABC≤log⁡n2n2.Moreover log⁡n-1=IG,ABC if and only if p=1 and q=n-1. i.e., G is a star. Also IG,ABC=log⁡(n/2n/2) if and only if p=n/2 and q=n/2; i.e., G is a complete bipartite graph (balanced).
Chemical graph is a graph 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 chemical graphs by taking ABC edge weights.Theorem 6.
Let G be a chemical graph withn vertices; then we have (16)log⁡ABC-log⁡116≤IG,ABC≤log⁡ABC-log⁡3Proof.
In a chemical graph G the maximum degree of a vertex is 4 and the minimum degree of a vertex is 1, so we have(17)IG,ABC=log⁡ABC-1/2ABC∑du+dv-2du.dvlog⁡du+dv-2du.dv=log⁡ABC-12ABC∑du+dv-2du.dvlog⁡du+dv-2-log⁡du.dv)≤log⁡ABC-12log⁡4+4-2-log⁡2≤log⁡ABC-log⁡3Similarly,(18)IG,ABC≥log⁡ABC-log⁡116Therefore,(19)log⁡ABC-log⁡116≤IG,ABC≤log⁡ABC-log⁡3Corollary 7.
LetG=(V,E,w) be any complete graph of order n. Then we have (20)IG,ABC≤nn-22-log⁡nn-1.Proof.
For any complete graph G of ordern we have [14], ABC(G)≤nn-2/2; therefore the result(21)IG,ABC≤nn-22-log⁡nn-1Corollary 8.
LetG=(V,E,w) be any tree of order n. Then we have (22)IG,ABC≤n-1n-2-log⁡nn-1.Corollary 9.
For a unicyclic graph(23)IG,ABC≤n2n2-7n-192n-1-log⁡nn-1.Corollary 10.
For a Star graph,(24)IG,ABC≤n2n2-7n-192n-1-log⁡nn-1.


## 3. Numerical Examples

Dendrimers are man-made, nanoscale compounds with unique properties that make them useful to the health and pharmaceutical industry as both enhancements to existing products and as entirely new products. Dendrimers are constructed by the successive addition of layers of branching groups. The final generation incorporates the surface molecules that give the dendrimers the desired function for pharmaceutical, life science, chemical, electronic, and materials applications. Dendrimers fall under the broad heading of nanotechnology, which covers the manipulation of matter in the size range of 1-100 nanometers (one million nanometers equal one millimeter) to create compounds, structures, and devices with a novel, predetermined properties.In this section, we discuss entropies of four familiar classes of dendrimers, namely, Porphyrin (Figure1), Propyl ether imine (Figure 2), Zinc-Porphyrin (Figure 3), and  P_oly(EThyleneAmidoAmine) (Figure 4) Dendrimers. It is important to remark that all dendrimers differ in cores. These dendrimers have been studied extensively in [25–28].Figure 1
Porphyrin dendrimer.Figure 2
Propyl ether imine dendrimer.Figure 3
Zinc-Porphyrin dendrimer.Figure 4
Poly(EThyleneAmidoAmine dendrimer.Example 1.
Let G be the Porphyrin dendrimers; then using the edge partition given of Porphyrin dendrimers given in Table1, we get(25)ABCG=77.26044062n-7.778174591Therefore(26)IG,ABC=log⁡ABC-1/2ABC∑(du+du-2du.dulog⁡du+du-2du.du=log⁡77.2604n-7.7781-1277.2604n-7.7781E163log⁡63+E232log⁡32+E322.log⁡22+E422.log⁡22+E523.log⁡23+E6156.log⁡156=log⁡77.2604n-7.7781-1277.2604n-7.77812n63log⁡63+24n32log⁡32+10n-522.log⁡22+48n-622.log⁡22+13n23.log⁡23+8n156.log⁡156=log⁡77.2604n-7.7781-1277.2604n-7.7781-23.308n+2.695Table 1
Edge partition of Porphyrin dendrimers based on degree of end vertices of each edge.
(du,dv) (1,3) (1,4) (2,2) (2,3) (3,3) (3,4) Number of edges 2n 24n 10n-5 48n-6 13n 8nExample 2.
Let G be the propyl ether imine dendrimer. Then using the edge partition of G given in Table2, we get(27)ABCG=0.7072n+1+0.7072n+4-16.970+4.2422nIG,ABC=log⁡ABC-12ABC∑(du+du-2du.dulog⁡du+du-2du.du=log⁡0.7072n+1+0.7072n+4-16.970+4.2422n-120.7072n+1+0.7072n+4-16.970+4.2422nE122log⁡22+E222log⁡22+E322log⁡22=log⁡0.7072n+1+0.7072n+4-16.970+4.2422n-120.7072n+1+0.7072n+4-16.970+4.2422n2n+122log⁡22+2n+422log⁡22]+6.2n-622log⁡22]=-1ln⁡0.70710678102.n+1.+0.70710678102.n+4.-16.97056275+0.70710678106.2n-0.24506453602.n+1.-0.24506453602.n+4.+5.881548863-0.24506453606.2nTable 2
Edge partition of propyl ether imine dendrimers based on degree of end vertices of each edge.
(du,dv) (1,2) (2,2) (2,3) Number of edges 2 n + 1 2n+4 6 . 2 n - 6Example 3.
For theZinc-Porphyrin dendrimers G, using edge partition given in Table 3, we get(28)
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Edge partition of Zinc-Porphyrin dendrimers based on degree of end vertices of each edge.
(du,dv) (2,2) (2,3) (3,3) (3, 4) Number of edges 16 .  2 n-4 40 .  2 n - 16 8 . 2 n - 16 4Example 4.
Let G be the graph ofP_oly(EThyleneAmidoAmine; then using edge partition given in Table 4 we get(29)ABCG=31.5502n-13.653IG,ABC=log⁡ABC-1/2ABC∑(du+du-2du.dulog⁡du+du-2du.du=log⁡31.5502n-13.653-1231.5502n-13.653E122log⁡22+E2(63log⁡63+E322log⁡22+E422log⁡22=log⁡44.9312n-22.226-1244.9312n-22.2264.2n22log⁡22]+4.2n-222log⁡22]+16.2n23log⁡23]+20.2n-9156log⁡156]=ln⁡31.550257562.n-13.65380844-1ln⁡31.550257562.n-13.65380844-10.464703192.n+4.497157986Table 4
Edge partition of Poly(EThyleneAmidoAmine dendrimers based on degree of end vertices of each edge.
(du,dv) (1,2) (1,3) (2,2) (2, 3) Number of edges 4 .  2 n 4 .  2 n - 2 16 . 2 n 20 .  2 n - 9Concluding Remarks. QSARs represent predictive models got from utilization of statistical instruments correlating biological activity (including desirable therapeutic effect and undesirable side effects) of chemicals (toxicants/drugs/environmental pollutants) with descriptors illustrative of molecular structure as well as properties. QSARs are being connected in many disciplines, for instance, toxicity prediction, risk assessment and regulatory decisions, lead optimization, and drug discovery. The atom-bond connectivity index denoted by ABC is a molecular structure descriptor that has remarkable application in rationalizing the stability of linear and branched alkanes and in the strain energy of cycloalkanes [25]. Weighted entropy is a generalization of Shannon’s entropy and is the measure of information supplied by a probablistic experiment whose elementary events are characterized both by their objective probabilities and by some qualitative (objective or subjective) weights [29]. It is useful to rank chemicals in quantitative high-throughput screening experiments [30] and may be used to balance the amount of information and the degree of homogeneity associated to a partition of data in classes [31]. Weighted entropy also found applications in the coding theory [32]. For more insights about applications of entropy, please see [33]. In this paper we have studied weighted entropy with atomic bond connectivity edge weights, which was an open problem of [34]. Our next aim is to work on entropy of weighted graphs with geometric arithmetic and sum connectivity edge weights.



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