On entropy measures of molecular graphs using topological indices

1 Introduction

Graph theory plays an important role in modeling and designing any chemical network. A large number of properties like physico-chemical properties, thermodynamic properties, chemical activity and biological activity are determined by the chemical applications of graph theory. These properties can be characterized by certain graph invariants referred to as topological indices. In chemical graph theory, molecular graph is a simple finite graph in which vertices denote the atoms and edges denote the chemical bonds in underlying chemical structure. The topological index is a numeric value or a sequence for a given structure under discussion, which actually shows the physical, chemical and biological properties of graph.

Chemical graph theory is a branch of mathematical chemistry in which tools of graph theory are applied to model the chemical phenomenon mathematically. Also, it relates to the nontrivial applications of graph theory for solving molecular problems. This theory contributes a prominent role in the field of chemical sciences, see for details (Gao et al., 2018). Also Chem-informatics is new subject which is a combination of chemistry, mathematics and information science. It examines Quantitative structure-activity relationship $(\mathit{QSAR})$ and Quantitative Structure-Property Relationship $(\mathit{QSPR})$ that are utilized to predict the bioactivity and physiochemical properties of chemical compounds (Wu et al., 2015).

Let G be a graph with $V(G)$ be the vertex set and $E(G)$ be the edge set of G. The degree $\xi (r)$ of a vertex r is the number of edges of G incident with vertex r. The length of a shortest path in a graph G is a distance $d(r,s)$ between r and s. If G is a graph with p vertices and q edges, then we say the order of G is p and the size of G is q. A graph of order p and size q is addressed as $(p,q)$ -graph.

In chemical graph, the vertices represent atoms and edges refer as the chemical bonds in underlying chemical structure. A topological index is a numerical value that is computed mathematically from the molecular graph. It is associated with chemical constitution indicating for correlation of chemical structure with many physical, chemical properties and biological activities. The exact formulas of topological indices for chemical graphs have been computed in (Siddiqui et al., 2016; Siddiqui et al., 2016). For further detail of application point see (Gao et al., 2017; Imran et al., 2018).

The concept of entropy was introduced first in Shannon’s famous paper (Shannon, 1948) as “ The entropy of a probability distribution is known as a measure of the unpredictability of information content or a measure of the uncertainty of a system”. Later, entropy was initiated to be applied to graphs nd chemical networks. It was developed for measuring the structural information of graphs and chemical networks. In 1955, Rashevsky, (Rashevsky, 1955) introduced the concept of graph entropy based on the classifications of vertex orbits. Recently, graph entropies have been widely applied in many different fields, such as chemistry, biology, ecology and sociology (Dehmer and Graber, 2013; Ulanowicz, 2004).

The graph entropy measures that associate probability distributions with elements (vertices, edges, etc.) of a graph can be classified as intrinsic and extrinsic measures. There are several different types of such graph entropy measures (Mowshowitz and Dehmer, 2012). The degree powers are extremely significant invariants and studied extensively in graph theory and network science, and they are used as the information functionals to explore the networks (Cao et al., 2014; Cao and Dehmer, 2015). Dehmer introduced graph entropies based on information functionals, which capture structural information, and studied their properties (Dehmer, 2008; Dehmer et al., 2012). For more expansive research, Estrada et.al proposed a physically-sound entropy measure for networks/graphs (Estrada and Hatano, 2007) and studied the walk-based graph entropies (Estrada, 2010).

2 Application of degree based entropies

Shannon’s seminal work (Shannon, 1948) in the late nineteen-forties marks the starting point of modern information theory. Following early applications in linguistics and electrical engineering, information theory was applied extensively in biology and chemistry, see, (Morowitz, 1953; Quastler, 1954). Here, the main novelty was the idea of considering a structure as an outcome of an arbitrary communication (Bonchev, 2003). With the aid of this insight, Shannon’s entropy formulas (Shannon, 1948) were used to determine the structural information content of a network (Sol and Valverde, 2004). As a result, this method has been used for exploring living systems, e.g., biological and chemical systems by means of graphs. These applications are closely related to the work of Rashevsky (1955) and Trucco (1956). In what follows, we review in chronological order graph entropy measures that have been used for studying biological and chemical networks.

Moreover entropy measures for graphs have been widely applied in biology, computer science and structural chemistry, see, (Dehmer and Mowshowitz, 2011). Broadly speaking, the applications for entopic network measures range from quantitative structure characterization in structural chemistry or software technology to explore biological or chemical properties of molecular graphs. We emphasize that the just mentioned applications relate to solve an underlying data analysis problem, e.g., a clustering or classification task. However, the so-called structural interpretation needs to be investigated as well. This calls to examine what kind of structural complexity does the measure detect. This problem is intricate as it is not clear on which graph class the measure should be evaluated. We conjecture that the introduced degree-based entropy can be used to measure network heterogeneity. Similar entopic measures which are based on vertex-degrees to detect network heterogeneity have been introduced by Sol and Valverde (2004) and Tan and Wu (2004).

3 Research aim and methodology

Our aim in this article, is to discuss the chemical graph of crystallographic structure of cuprite ${\mathit{Cu}}_{2}O[m,n,t]$ and crystal structure of titanium difluoride ${\mathit{TiF}}_2[m,n,t]$ . Also, we compute entropies of these structures by making a relation of degree based topological indices namely first redefined Zagreb index, second redefined Zagreb index, third redefined Zagreb index, fourth Atom Bond Connectivity index, fifth Geometric Arithmetic index, Sanskruti index with the help of information function.

The methodology of this article is as follows. In Section 4, we introduce the definition of degree based topological indices of graph. In Section 5, we introduce the definition of degree based and edge weight based entropy of Graph. In Section 6, we discuss the crystallographic structure of Molecule ${\mathit{Cu}}_{2}O[m,n,t]$ and compute the corresponding entropies. In Section 7, we discuss the crystal structure of titanium difluoride ${\mathit{TiF}}_2[m,n,t]$ and compute the corresponding entropies. In Section 8, we present the comparisons of these computes results for ${\mathit{Cu}}_{2}O$ . In Section 9, we present the comparisons of these computes results for ${\mathit{TiF}}_2[m,n,t]$ . In Section 10, we represent the conclusion of this paper.

4 Degree based topological indices of graph

The modified version of the Zagreb indices were introduced by Ranjini et al. (2013), namely, the redefined first, second and third Zagreb index for a graph G as; $$\begin{array}{cc}\hfill {\mathit{ReZG}}_1(G)=& {\displaystyle \sum _{\mathit{rs}\in E(G)}}\frac{\xi (r)+\xi (s)}{\xi (r)\xi (s)}\hfill \\ \hfill {\mathit{ReZG}}_2(G)=& {\displaystyle \sum _{\mathit{rs}\in E(G)}}\frac{\xi (r)\xi (s)}{\xi (r)+\xi (s)}\hfill \\ \hfill {\mathit{ReZG}}_3(G)=& {\displaystyle \sum _{\mathit{rs}\in E(G)}}(\xi (r)\xi (s))(\xi (r)+\xi (s))\hfill \end{array}$$ The fourth version of atom bond connectivity index ${\mathit{ABC}}_4$ of a graph G is introduced by Ghorbani and Hosseinzadeh (2010): $${\mathit{ABC}}_4(G)={\displaystyle \sum _{\mathit{rs}\in E(G)}}\sqrt{\frac{S_r+S_s-2}{S_r{S}_s}}$$ where $S_r={\sum }_{\mathit{rs}\in E(G)}\xi (s)$ and $S_s={\sum }_{\mathit{rs}\in E(G)}\xi (r)$ .

The fifth version of geometric arithmetic index ${\mathit{GA}}_5$ of a graph G is introduced by Graovac et al. (2011) as: $${\mathit{GA}}_5(G)={\displaystyle \sum _{\mathit{rs}\in E(G)}}\frac{2\sqrt{S_r{S}_s}}{S_r+S_s}$$ In 2016, Hosamani (2016) introduced the Sanskruti index $S(G)$ for a molecular graph G as: $$S(G)={\displaystyle \sum _{\mathit{rs}\in E(G)}}{\left(\frac{S_r\times S_s}{S_r+S_s-2}\right)}^3$$

5 Degree based and edge weight based entropy of graph

In 2014, Chen et al. (2014a) introduced the definition of the entropy of edge weighted graph. For an edge weighted graph, $G=(V(G)\text{;}E(G)\text{;}\psi (\mathit{rs}))$ , where $V(G)$ , $E(G)$ and $\psi (\mathit{rs})$ denote the vertex set, the edge set and the edge weight of edge $(\mathit{rs})$ , respectively. Then the entropy of edge weighted graph is represented in Eq. (1).

• The Redefined first Zagreb Entropy

If $\psi (\mathit{rs})=\frac{\xi (r)+\xi (s)}{\xi (r)\xi (s)}$ , then $${\displaystyle \sum _{\mathit{rs}\in E(G)}}\psi (\mathit{rs})={\displaystyle \sum _{\mathit{rs}\in E(G)}}\frac{\xi (r)+\xi (s)}{\xi (r)\xi (s)}={\mathit{ReZG}}_1(G)$$ Now Eq. (1) reduced in the following from and is called The Redefined first Zagreb Entropy.

• The Redefined Second Zagreb Entropy

If $\psi (\mathit{rs})=\frac{\xi (r)\xi (s)}{\xi (r)+\xi (s)}$ , then $${\displaystyle \sum _{\mathit{rs}\in E(G)}}\psi (\mathit{rs})={\displaystyle \sum _{\mathit{rs}\in E(G)}}\frac{\xi (r)\xi (s)}{\xi (r)+\xi (s)}={\mathit{ReZG}}_2(G)$$ Now Eq. (1) reduced in the following from and is called The redefined second Zagreb Entropy.

• The Redefined Third Zagreb Entropy

If $\psi (\mathit{rs})=\left(\xi (r)\xi (s)\right)\left(\xi (r)+\xi (s)\right)$ , then $${\displaystyle \sum _{\mathit{rs}\in E(G)}}\psi (\mathit{rs})={\displaystyle \sum _{\mathit{rs}\in E(G)}}\left(\xi (r)\xi (s)\right)\left(\xi (r)+\xi (s)\right)={\mathit{ReZG}}_3(G)$$ Now Eq. (1) reduced in the following from and is called The redefined third Zagreb Entropy.

• The Fourth Atom Bond Connectivity Entropy

If $\psi (\mathit{rs})=\sqrt{\frac{S_r+S_s-2}{S_r{S}_s}}$ , then $${\displaystyle \sum _{\mathit{rs}\in E(G)}}\psi (\mathit{rs})={\displaystyle \sum _{\mathit{rs}\in E(G)}}\sqrt{\frac{S_r+S_s-2}{S_r{S}_s}}={\mathit{ABC}}_4(G)$$ Now Eq. (1) reduced in the following from and is called The Fourth Atom Bond Connectivity Entropy.

• The Fifth Geometric Arithmetic Entropy

If $\psi (\mathit{rs})=\frac{2\sqrt{S_r{S}_s}}{S_r+S_s}$ , then $${\displaystyle \sum _{\mathit{rs}\in E(G)}}\psi (\mathit{rs})={\displaystyle \sum _{\mathit{rs}\in E(G)}}\frac{2\sqrt{S_r{S}_s}}{S_r+S_s}={\mathit{GA}}_5(G)$$ Now Eq. (1) reduced in the following from and is called the Fifth Geometric Arithmetic Entropy.

• The Sanskruti Entropy

6 Crystallographic structure of ${\mathit{Cu}}_{2}O[m,n,t]$

Among various transition metal oxides, ${\mathit{Cu}}_{2}O$ has attracted large attention in recent years owing to its distinguished properties and non-toxic nature, low-cost, abundance, and simple fabrication process (Chen et al., 2014b). Nowadays, the promising applications of ${\mathit{Cu}}_{2}O$ mainly focus on chemical sensors, solar cells, photocatalysis, lithium-ion batteries and catalysis (Yuhas and Yang, 2009). The chemical graph of Crystallographic structure of ${\mathit{Cu}}_{2}O$ described in Fig. 1 and Fig. 2, see details in (Zhang et al., 2006). Let $G\cong {\mathit{Cu}}_{2}O[m,n,t]$ be the chemical graph of ${\mathit{Cu}}_{2}O$ with $m\times n$ unit cells in the plane and t layers. We construct this graph first by taking $m\times n$ unites in the $\mathit{mn}$ -plane and then storing it up in t layers. The number of vertices and edges of ${\mathit{Cu}}_{2}O[m,n,t]$ are $(m+1)(n+1)(t+1)+5\mathit{mnt}$ and $8\mathit{mnt}$ , respectively.

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Fig. 1

Crystallographic structure of Molecule ${\mathit{Cu}}_{2}O$ . (a) Structural characteristics of Cu and O atoms in the ${\mathit{Cu}}_{2}O$ lattice. The ${\mathit{Cu}}_{2}O$ lattice is formed by interpenetrating the Cu and O lattices with each other. (b) Unit cell of ${\mathit{Cu}}_{2}O$ . Copper atoms are shown as small blue spheres, and oxygen atoms are shown as large red spheres. In the ${\mathit{Cu}}_{2}O$ lattice, each Cu atom is coordinated with two O atoms, and each O atom is coordinated with four Cu atoms.

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Fig. 2

Crystallographic structure of ${\mathit{Cu}}_{2}O[3,2,3]$ .

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To compute our result we make the partitions of the vertices and edge. More preciously the vertex partition of ${\mathit{Cu}}_{2}O[m,n,t]$ based on degrees of each vertex is depicted in Table 1. Also the edge partition of ${\mathit{Cu}}_{2}O[m,n,t]$ based on degrees of end vertices of each edge are depicted in Table 2.