# Irregularity Measures of Subdivision Vertex-Edge Join of Graphs

Authors: Jialin Zheng; Shehnaz Akhter; Zahid Iqbal; Muhammad Kashif Shafiq; Adnan Aslam; Muhammad Ishaq; Muhammad Aamir
Journal: Journal of Chemistry
 (2021)

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## Abstract

The study of graphs and networks accomplished by topological measures plays an applicable task to obtain their hidden topologies. This procedure has been greatly used in cheminformatics, bioinformatics, and biomedicine, where estimations based on graph invariants have been made available for effectively communicating with the different challenging tasks. Irregularity measures are mostly used for the characterization of the nonregular graphs. In several applications and problems in various areas of research like material engineering and chemistry, it is helpful to be well-informed about the irregularity of the underline structure. Furthermore, the irregularity indices of graphs are not only suitable for quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) studies but also for a number of chemical and physical properties, including toxicity, enthalpy of vaporization, resistance, boiling and melting points, and entropy. In this article, we compute the irregularity measures including the variance of vertex degrees, the total irregularity index, theσ irregularity index, and the Gini index of a new graph operation.


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## Body

## 1. Introduction

Mathematical chemistry is showing assistance by providing the time-saving and competent tools for the characterizations of chemical compounds instead of any other data as input. Topological invariants catch some key topological and combinatorial details of the underline structures, and with the help of these, some properties of these structures can be deduced without the assistance of the quantum mechanics [1–3]. Topological invariants are numerical quantities that characterize various structural properties of an underline structure. In pharmacy, mathematical chemistry, and environmental sciences, these descriptors are used for the QSAR/QSPR studies, where biological or chemical activities and physical properties of an underline structure are associated with its molecular structure. Therefore, these descriptors are mostly quoted as molecular descriptors [4]. Furthermore, topological descriptors are becoming essential also in the theory of networks and complex systems [5]. More precisely, they are practised for quantifying numerous global and local properties in biological networks, social networks, neural networks, and communications networks. As a result, topological descriptors can be exercised in different scientific fields. However, molecular descriptors generally require to be computed of various families of molecules, and hence, algorithms and effective methods for the computation are required. Therefore, determining these numerical indices is one of the good-looking areas of research. Some well-known classes of numerical descriptors are distance-based, degree-based, and counting-related. Degree-based indices are the most frequently used invariants these days [4, 6–9]. Some correlations of the topological indices with the chemical reactivity, physical properties, or biological activity of compounds can be seen in [2, 3, 6–9]. In short, these invariants take into account the internal atomic arrangement of the compounds and yield several facts in the mathematical form about the multiple bonds, presence of heteroatoms, molecular size, branching, and shape. Among degree-based indices, irregularity measures may play an important role in chemistry, especially in the QSPR/QSAR studies [1, 10] as well as network theory [5, 11]. These measures have been chosen to analyse the topological aspects of the underline structures; for instance, see [5, 10–17].In 1957, Collatz and Sinogowitz introduced the term “network irregularity” [18]. In fact, irregularity measures are used in complex networks, which speak on behalf of disparate systems. These systems have main topological features like self-similarity, scale-freeness, network motifs, and small worldness [5]. On the contrary, the molecular complexity has been discussed in the scientific literature from decades; due to its ever-increasing significance, one is interested to go through the process of estimating the degree of complexity. One way is to work out the irregularity measures of the underlined structures [13, 15, 19, 20]. By using irregularity measures, Reti et al. examine the physicochemical properties, including an acentric factor of octane isomers entropy, the entropy of vaporization, boiling point, and standard enthalpy of vaporization [10]. Actually, they showed that, by using these irregularity measures, some properties of isomers can be estimated with excellent accuracy. Estrada examined the irregularity of various networks in distinct organisms and showed that these networks achieve higher irregularity as compared to some regular networks [11]. So, these facts provide motivation to study the irregularity of structures.A graph will be a regular graph if all its vertices have the same degrees. If not, then it will be an irregular graph. Recently, the total irregularity index has been suggested, and concerning this index, some well-known graphs have been studied. Bell examined the irregularity of graphs by using different irregularity indices [13]. Gutman provided some basic results for some irregularity measures of graphs in [15].Throughout this paper, all graphs will assume to be simple, finite, and connected. We represent the vertex set of a graphQ with VQ and edge set with EQ. The order and size of Q are presented as p and q, respectively. The degree of a vertex q∈VQ, denoted as μQq, is the number of linked vertices to q in Q. A topological index is said to be an irregularity measure if this has a nonzero value for a nonregular graph, and it has a value equal to zero for a regular graph. The applications of regular graphs in chemical graph theory appeared after the discovery of fullerenes and nanotubes. The easiest manners of manifesting the irregularities are by using irregularity measures. Most of the irregularity measures used the idea of the imbalance of an edge e=qq′∈EQ specified as imbe:=μQq−μQq′. In 1997, the term “The graph irregularity” was proposed by Albertson [19]. For a graph Q, it is symbolized by irrQ and detailed as follows:(1)irrQ=∑e∈EQimbe.This invariant is also familiar as the third Zagreb index. Albertson [19] presented some upper bounds of irregularity for trees and bipartite and triangle-free graphs. The connections between the irregularity of trees and unicyclic graphs with their respective matching numbers were explored in [21]. Hansen and Melot [22] described the graphs having the maximal irregularity index. Abdo and Dimitrov [23] computed the irregularity indices of some graph operations. The total irregularity index of a graph Q was introduced in [24] in the following way:(2)irrtQ=12∑q,q′∈VQμQq−μQq′.The characterization of graphs with respect to extremal irregularity and the relationships among the various irregularity measures are studied in [25]. By using the irregularity of graphs, Fath-Tabar [26] provided novel bounds of Zagreb indices. For the comprehensive study about these irregularity measures, we refer [27, 28]. The σ irregularity index of a graph Q is introduced by Gutman et al. in [29] as follows:(3)σQ=∑qq′∈EQμQq−μQq′2.For different properties of this index, the interested reader may refer to [30, 31]. If the size and order of Q are q and p, then the variance of Q can be expressed as follows [13]:(4)VarQ=1p∑q∈VQμQ2Q−1p2∑q∈VQμQQ2.Gini index ofQ is denoted by ζQ and is defined as(5)ζQ=12pqirrtQ.In 1972, Gutman and Trinajestic [1] the first and second Zagreb indices as follows:(6)M1Q=∑q∈VQμQ2q,M2Q=∑qq′∈EQμQqμQq′.These indices have significant applications in chemistry and its related fields. The forgotten topological index ofQ was initiated by Furtula and Gutman in [32] in the following way:(7)FQ=∑q∈VQμQ3q.Like Zagreb indices, the firstIRM1 and the second IRM2 irregularity indices are as follows [33]:(8)IRM1Q=1p∑q∈VQηQq−2qp,IRM2Q=∑q∈VQμQq−ηQq,where ηQq is the average of the degrees of the adjacent vertices of q.A novel graph can be designed from the given graphs by using graph operations, and it is demonstrated that a number of chemical graphs can be constructed with the help of graph operations. The relations achieved for different topological descriptors of graph operations in the form of topological descriptors of their components; thus, it is valuable to find out the topological invariants of some molecular graphs and nanostructures. There are a number of studies about the topological indices of various graph operations; for instance, see [34–50]. In short space of time, novel graph operation, famous as the subdivision vertex-edge join (SVE-join), is suggested in [51]. For a graph Q, let SQ represents the subdivision graph of it, and the vertex set of SQ can be divided into two parts: the first one is the VQ, and the second one, represented by ℐQ, that contains the inserting vertices is related to the edges of Q. Let us consider two disjoint graphs T and Y. The SVE-join of Q with T and Y, symbolized by QS⊳TV∪YE, is the graph consisting of SQ, T and Y, all vertex-disjoint, and joining the i-th vertex of VQ to each vertex of VT and i-th vertex of ℐQ to each vertex of VY. For example, see Figure 1. It is easy to see that QS⊳TV∪YE has order p1+q1+p2+p3 and its size is 2q1+p1p2+q1p3+q2+q3, where p1 and q1 are the number of vertices and edges of Q, p2 and q2 are the order and size of T, and p3 and q3 are the order and size of Y. Furthermore, we see that QS⊳TV∪YE is Q∨˙T if Y is the null graph and is Q∨¯Yif T is the null graph [52].Figure 1
C4∨˙K4, C4∨¯C3, and C4S⊳K4V∪C3E.
(a)(b)(c)(d)(e)(f)


## 2. Main Results

In this section, first, by using the definition of SVE-join, we express several properties of the subdivision vertex-edge join of three graphs. After this, by applying these consequences, we provide our main results.Lemma 1.
For the graphsQ, T, and Y, we have the following:(1)
μQS⊳TV∪YEq=μTq+p1ifq∈VTμYq+q1ifq∈VYμQq+p2ifq∈VQp3+2ifq∈EQ(2)
ηQS⊳TV∪YEq=μTq+p1ifq∈VTμYq+q1ifq∈VYμQq+p2ifq∈VQp3+2ifq∈EQHere,ωTq is the sum of degrees of its linked vertices in T and ωYq is the sum of degrees of its linked vertices in Y.Next, we give the expression of the variance of the SVE-join of three graphs.Theorem 1.
For the graphsQ, T, and Y, we have the following:(9)VarQS⊳TV∪YE=1p1+p2+p3+q1M1Q+M1T+M1Y+p1p2p1+p2+4p2q1+p1q2+p3q1q1+p3+4q11+q3+p3−22q1+q2+q3+p1p2+p3q1p1+p2+p3+q1.Proof.
By using Lemma1 in (4), we obtain(10)VarQS⊳TV∪YE=1p1+p2+p3+q1∑q∈VQS⊳TV∪YEμQS⊳TV∪YE2Q−1p1+p2+p3+q12∑q∈VQS⊳TV∪YEμQS⊳TV∪YEQ2=1p1+p2+p2+q1∑q∈VQμQQ+p22+∑q∈EQp3+22+∑q∈VTμTQ+p12+∑q∈VYμYQ+q12+1p1+p2+p3+q12∑q∈VQμQQ+p2+∑q∈EQp3+2+∑q∈VTμTQ+p1+∑q∈VYμYQ+q12=1p1+p2+p3+q1M1Q+p22p1+4p2q1+q1p32+4+4p3+M1T+p12p2+4p1q2+M1Y+q12p3+4q1q3+1p1+p2+p3+q122q1+p1p2+q1p3+2+2q2+p1p2+2q3+q1p3=1p1+p2+p3+q1M1Q+M1T+M1Y+p1p2p1+p2+4p2q1+p1q2+p3q1q1+p3+4q11+q3+p3−22q1+q2+q3+p1p2+p3q1p1+p2+p3+q1.
This completes the proof.We proceed further to construct an upper bound of the total irregularity index of the SVE-join of three graphs in terms of their orders and sizes.Theorem 2.
For the graphsQ, T, and Y, we have the following:(11)irrtQS⊳TV∪YE≤irrtQ+irrtT+irrtY+p1p22p2−p1+p1p3p2+q1+p2q1p1−p3+p3q12p3−q1+q1p2+q1−p1+2p3+q2p1+q1+p3+q3q1+p2+p1.Proof.
By using Lemma1 in (2), its follows that(12)irrtQS⊳TV∪YE=12∑q,q′∈VQS⊳TV∪YEμQS⊳TV∪YEq−μQS⊳TV∪YEq′=12∑q,q′∈VTμTq+p1−μTqt′−p1+12∑q,q′∈VYμYq+q1−μYq′−q1+12∑q,q′∈VQμQq+p2−μQq′−p2+12∑q,q′∈EQp3+2−p3−2+12∑q∈VQ∑q′∈VTμQq+p2−μTq′−p1+12∑q∈VQ∑q′∈VYμQq+p2−μYq′−q1+12∑q∈VQ∑q′∈EQμQq+p2−p3−2+12∑q∈VT∑q′∈EQμTq+q1−p3−2+12∑q∈VT∑q′∈VYμTq+p1−μYq′−q1+12∑q∈EQ∑q′∈VYp3+2−μYq′−q1≤irrtQ+irrtT+irrtY+0+12∑q∈VQ∑q′∈VTμQq+12∑q∈VQ∑q′∈VTμTq′+12∑q∈VQ∑q′∈VTp2−p1+12∑q∈VQ∑q′∈VYμQq+12∑q∈VQ∑q′∈VYμYq′+12∑q∈VQ∑q′∈VYp2−q1+12∑q∈VQ∑q′∈EQμQq+12∑q∈VQ∑q′∈EQp2−p3−2+12∑q∈VT∑q′∈EQμTq+12∑q∈VT∑q′∈EQq1−p3−2+12∑q∈VT∑q′∈VYμTq+12∑q∈VT∑q′∈VYμYq′+12∑q∈VT∑q′∈VYp1−q1+12∑q∈EQ∑q′∈VYμYq′+12∑q∈EQ∑q′∈VYp3+2−q1=irrtQ+irrtT+irrtY+p2q1+p1q2+p1p22p2−p1+q1p3+p1q3+p1p32p2−q1+q12+p1q12p2−p3−2+q1p2+q1p22p1−p3−2+q1q2+p2p32p1−q1+q2p3+q3p2+q1p32p3+2−q1+q1q3.
Result follows from simplifications.
In the upcoming corollary, we give an upper bound of Gini index ofQS⊳TV∪YE. The proof follows from Theorem 2 and equation (5) and hence omitted.Corollary 1.
For the graphsQ, T, and Y, we have(13)ζQS⊳TV∪YE≤12p1+p2+p3+q12q1+q2+q3+p1p2+p3q1irrtQ+irrtT+irrtY+p1p22p2−p1+p1p3p2+q1+p2q1p1−p3+p3q12p3−q1+q1p2+q1−p1+2p3+q2p1+q1+p3+q3q1+p2+p1.In the next result, we find the formula ofσ index for the subdivision vertex-edge join of three graphs in terms of their orders and sizes.Theorem 3.
For the graphsQ, T, and Y, we have(14)σQS⊳TV∪YE=σT+σY+FQ+3p2−2p3−4M1Q+p1M1T+q1M1Y+p1p2p2−p12−8q1q2+4p2−p1p2q1−p1q2−4q1q3p3−q1+2+2q1p33p3−2q1+6+q1p3p3−q12+2q1p2−22−4p2p3q1.Proof.
By using Lemma1 in (3), it follows that(15)σQS⊳TV∪YE=∑qq′∈EQS⊳TV∪YEμQS⊳TV∪YEQ−μQS⊳TV∪YEq′2=∑qq′∈ETμTQ+p1−μTq′−p12+∑qq′∈EYμYQ+q1−μYq′−q12+∑q∈VQ∑q′∈VTμQQ+p2−μTq′−p12+∑q∈EQ∑q′∈VYp3+2−μYq′−q12+∑q∈VQ∑q′∈EQμQQ+p2−p3−22=∑qq′∈ETμTQ−μTq′2+∑qq′∈EYμYQ−μYq′2+∑q∈VQ∑q′∈VTμQ2Q+μT2q′+p2−p12−2μQQμTq′+2p2−p1μQQ−2p2−p1μTq′+∑q∈EQ∑q′∈VYp3+2−q12+μY2q′−2p3+2−q1μYq′+∑q∈VQμQQμQ2Q+p2−p3−22+2p2−p3−2μQQ=σT+σY+∑q∈VQ∑q′∈VTμQ2Q+∑q∈VQ∑q′∈VTμQ2Q+p1p2p2−p12−2∑q∈VQμQQ∑q′∈VTμTq′+2p2−p1∑q∈VQ∑q′∈VTμQQ−2p2−p1∑q∈VQ∑q′∈VTμTq′+q1p3p3+2−q12+∑q∈EQ∑q′∈VYμY2q′−2p3+2−q1∑qq∈EQ∑q′∈VYμYq′+∑q∈VQμQ3Q+p2−p3−22∑q∈VQμQQ+2p2−p3−2∑q∈VQμQ2Q=σT+σY+p2M1Q+p1M1T+p1p2p2−p12−22q12q2+2p2−p12q1p2−2p2−p12q2p1+q1p3p3+2−q12+q1M1Y−2p3+2−q1q12q3+FQ+2q1p2−p3−22+2p2−p3−2M1Q.
Result follows after some simplifications.
Next, we determine the expression for theIRM1 index of QS⊳TV∪YE.Theorem 4.
LetQ, T, and Y be the three graphs. Then,(16)IRM1QS⊳TV∪YE≤ΓT+ΓY+2q2+p1p2IDQ+2q1+p1p2IDT+q1p3+2IDY+1p3+2M1Q−p1p2−q1p3p3+2−q1p3+2+2q1p2+q3p3+2+p1p3+2−22q1+q2+q3,where ΓT=∑q∈VTωTq/μTq and ΓY=∑q∈VYωYq/μYq.Proof.
By using Lemma1 in (8), we obtain(17)IRM1QS⊳TV∪YE=1p1+p2+p3+q1∑q∈VQS⊳TV∪YEηQS⊳TV∪YEQ−22q1+p1p2+q1p3+q2+q3p1+p2+p3+q1=1p1+p2+p3+q1∑q∈VTωTQ+p1μTQ+2q1+p1p2μTQ+p1+∑q∈VYωYQ+q1μYQ+q1p3+2μYQ+q1+∑q∈VQμQQp3+2+2q2+p1p2μQQ+p2+∑q=qq′∈EQμQQ+μQq′+2p2+2q3+q1p3p3+2−22q1+p1p2+q1p3+q2+q3p1+p2+p3+q1.
Since1/μTq+p1≤1/μTq, 1/μYq+q1≤1/μYq and 1/μQq+p2≤1/μQq. Therefore,(18)IRM1QS⊳TV∪YE≤1p1+p2+p3+q1∑q∈VTωTQμTQ+p1+2q1+p1p21μTQ+∑q∈VYωYQμYQ+q1+q1p3+21μYQ+∑q∈VQp3+2+2q2+p1p21μQQ+1p3+2∑q=qq′∈EQμQQ+μQq′+2p2+2q3+q1p3−22q1+p1p2+q1p3+q2+q3p1+p2+p3+q1=1p1+p2+p3+q1∑q∈VTωTQμTQ+∑q∈VTp1+2q1+p1p2∑q∈VT1μTQ+∑q∈VYωYQμYQ+∑q∈VYq1+q1p3+2∑q∈VY1μYQ+∑q∈VQp3+2+2q2+p1p2∑q∈VQ1μQQ+1p3+2∑q=qq′∈EQμQQ+μQq′+1p3+2∑q=qq′∈EQ2p2+2q3+q1p3−22q1+p1p2+q1p3+q2+q3p1+p2+p3+q1=1p1+p2+p3+q1ΓT+p1p2+2q1+p1p2IDT+ΓY+p3q1+q1p3+2IDY+p1p3+2+2q2+p1p2IDQ+1p3+2M1Q+q12p2+2q3+q1p3p3+2−4q1−2p1p2−2q1p3−2q2−2q3=1p1+p2+p3+q1ΓT+ΓY+2q2+p1p2IDQ+2q1+p1p2IDT+q1p3+2IDY+1p3+2M1Q−p1p2−p3q11−q1p3+2+2q1p2+q3p3+2+p1p3+2−22q1−q2−q3.
Result follows after some simplifications.
Finally, we give an upper bound for theIRM2 index of the subdivision vertex-edge join of Q, T, and Y.Theorem 5.
For the graphsQ, T, and Y, we have(19)IRM2QS⊳TV∪YE≤ΓT+ΓY+2q1+p1p2IDT+q1p3+2IDY+2q2+p1p2IDQ+1p3+2M1Q+2q1+q2+q3+3p1p2+q1p32p3+2+q1p3+1+p1+q1p3+2+2q1p2+q3p3+2,where ΓT=∑q∈VTωTq/μTq and ΓY=∑q∈VYωYq/μYq.Proof.
By using Lemma1 in (8), we obtain(20)IRM2QS⊳TV∪YE=∑q∈VQS⊳TV∪YEμQS⊳TV∪YEq−ηQS⊳TV∪YEq=∑q∈VTμTq+p1−ωTq+p1μTq+2q1+p1p2μTq+p1+∑q∈VYμYq+q1−ωYq+q1μYq+q1p3+2μYq+q1+∑q∈VQμQq+p2−μQqp3+2+2q2+p1p2μQq+p2+∑q=qq′∈EQp3+2−μQq+μQq′+2p2+2q3+q1p3p3+2.
Since for any real numbersa and b, we have a−b≤a+b. Therefore,(21)IRM2QS⊳TV∪YE≤∑q∈VTμTq+p1+∑q∈VYμYq+q1+∑q∈VQμQq+p2+∑q=qq′∈EQp3+2+∑q∈VTωTq+p1μTq+2q1+p1p2μTq+p1+∑q∈VYωYq+q1μYq+q1p3+2μYq+q1+∑q∈VQμQqp3+2+2q2+p1p2μQq+p2+∑q=qq′′∈EQμQq+μQq′+2p2+2q3+q1p3p3+2.
Since1/μTq+p1≤1/μTq, 1/μYq+q1≤1/μYq and 1/μQq+p2≤1/μQq:(22)IRM2QS⊳TV∪YE≤∑q∈VTμTQ+p1+∑q∈VYμYQ+q1+∑q∈VQμQQ+p2+∑q=qq′∈EQp3+2+∑q∈VTωTQμTQ+p1+2q1+p1p21μTQ+∑q∈VYωYQμYQ+q1+q1p3+21μYQ+∑q∈VQp3+2+2q2+p1p21μQQ+1p3+2∑q=qq′∈EQμQQ+μQq′+2p2+2q3+q1p3=2q2+p1p2+2q3+q1p3+2q1+p1p2+q1p3+2+ΓT+p1p2+2q1+p1p2IDT+ΓY+q1p3+q1p3+2IDY+p1p3+2+2q2+p1p2IDQ+1p3+2M1Q+q1p3+22p2+2q3+q1p3=ΓT+ΓY+2q1+p1p2IDT+q1p3+2IDY+2q2+p1p2IDQ+1p3+2M1Q+2q1+q2+q3+3p1p2+q1p32+q1p3+1+p1+q1p3+2+2q1p2+q3p3+2.
The required result follows after some simplifications.


## 3. Conclusions

It is important to emphasis that as a large number of graphs are composed by smaller ones using graph products (and, as an outcome, their properties are strongly interconnected), the study of graph products is a timely and suitable research subject. Furthermore, the analysis of networks and graphs through their graph descriptors succeeds in playing an appreciable role to understand their underlying topologies, as it is commonly applied also in cheminformatics, bioinformatics, and biomedicine, where assessments derived through these invariants are valid for effectively communicating with the different hard tasks. Hence, the irregularity measures of connected graphs are manifested by a mathematical derivation approach. Our outcomes could perform a meaningful task in assessing the properties of underline structures.



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