# On Ve-Degree-Based Irregularity Properties of the Crystallographic Structure of Molecules

Authors: Akbar Jahanbani; Rana Khoeilar; Hajar Shooshtari
Journal: Journal of Chemistry
 (2021)

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## Abstract

Irregularity indices are usually used for quantitative characterization of the topological structure of nonregular graphs. In numerous applications and problems in material engineering and chemistry, it is useful to be aware that how irregular a molecular structure is? In this paper, we are interested in formulating closed forms of irregularity measures of some of the crystallographic structures ofCu2Op,q,r and crystallographic structure of titanium difluoride of TiF2p,q,r. These theoretical conclusions provide practical guiding significance for pharmaceutical engineering and complex network and quantify the degree of folding of long organic molecules.


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## Body

## 1. Introduction

In the medicines mathematical model, the structure of medication is taken as an undirected graph, where vertices and edges are taken as atoms and chemical bonds. Mathematical chemistry gives tools, for example, polynomials and numbers, to obtain properties of chemical compounds without utilizing quantum mechanics [1–3]. A topological index is a numerical parameter of a graph and describes its topology. It depicts the molecular structure numerically and is utilized in the advancement of qualitative structure-activity relationships (QSARs). There are three kinds of topological indices:(1)
Degree-based.(2)
Distance-based.(3)
Spectral-based.In theoretical chemistry and biology, topological indices have been used for working out the information on molecules in the form of numerical coding. This relates to characterizing physicochemical, biological, toxicologic, pharmacologic, and other properties of chemical compounds. Thousands of molecular structure descriptors have been suggested in order to characterize the physical and chemical properties of molecules [4–6].Degree-based indices can be further classified in the class of irregularity indices that measure the irregularity of the given graph. Recently, Réti et al. [7, 8] showed that the graph irregularity indices are efficient in quantitative structure-property relationship (QSPR) studies of molecular graphs [9].LetCu2Op,q,r be the chemical graph of Cu2O with p×q unit cells in the plane and r layers. Copper(I) oxide or cuprous oxide is the inorganic compound with the formula Cu2O. Figure 1 describes the graph of the molecule Cu2O [10]. Note that copper atoms are shown in Figure 1 by red dots and oxygen atoms are shown by blue dots. In the Cu2O lattice graph, each copper atom is attached to two oxygen atoms, and every oxygen atom is attached to four copper atoms.Figure 1
The structure of the moleculeCu2O.
(a)(b)It is one of the principal oxides of copper, the other beingCuO or cupric oxide. Nowadays, the crystallographic structure of the molecule Cu2O has attracted attention due to its interesting properties, low-cost, abundance, nontoxic nature [11]. This is the main reason to choose Cu2O and compute irregularity indices for it. Figures 1–5 represent molecular graphs of these two systems.Figure 2
The graph ofCu2O1,1,1.Figure 3
The graph ofCu2O3,2,3.Figure 4
The graph ofTiF21,1,1.Figure 5
The graph ofTiF24,1,2.Zahid et al. [12] computed the irregularity indices of a nanotube, Gao et al. [13] recently computed irregularity measures of some dendrimer structures, in [14], they had discussed irregularity molecular descriptors of hourglass, jagged-rectangle, and triangular benzenoid systems, Iqbal et al. [15] computed the irregularity indices of nanosheets, Zheng et al. [16] discussed irregularity measures of subdivision vertex-edge, Abdo et al. [17] computed irregularity of some molecular structures, Iqbal [12] studied irregularity measures of some nanotubes, and Gao et al. [18] obtained M-polynomials of the crystallographic structure of molecules. In [19], the authors investigated the total irregularity of trees with bounded maximal degree Δ and state integer linear programming problem which gives standard information about extremal trees. In many applications and problems, it is of importance to know how much a given graph deviates from being regular, i.e., how great its irregularity is [20]. Since then, irregular graphs and the degree of irregularity have become one of the core open problems of graph theory. In the current article, we are interested in finding the irregularity of the crystallographic structure of molecules Cu2O and TiF2p,q,r and computing and comparing the irregularities of some relevant chemical graphs.Throughout this article, all graphs are finite, undirected, and simple. LetG=VG,EG be such a graph with vertex set VG and edge set EG. The order of G is the cardinality of its vertex set, and size is the cardinality of its edge set. The vertices of G correspond to atoms, and an edge between two vertices is related to the chemical bond between these vertices. The degree of a vertex u of a graph G is symbolized by dGu=du and is defined as the number of edges incident with u. A graph is said to be regular if all its vertices have the same degree; otherwise, it is irregular. For details on the bases of graph theory, we refer to the book [21].For a graphG, Albertson [22] defines the imbalance of an edge e=uv∈EG as dGu−dGv and the irregularity of G as(1)IRR=IRRG=∑uv∈EGdu−dv.For more information about theIRRG, you can see [23]. Abdo et al. [24] introduced the total irregularity index, as follows:(2)IRRt=IRRtG=12∑uv∈EGdu−dv.Gutman et al. [25] introduced the IRFG irregularity index of the graph G as follows:(3)IRF=IRFG=∑uv∈EGdu−dv2.Simplified ways of expressing the irregularities are irregularity indices. These irregularity indices have been studied recently in a novel way [26, 27]. The first such irregularity index was introduced in [28]. Table 1 shows the rest of the irregularity indices used in this paper.Table 1
Some irregularity indices considered in [8].
Irregularity indicesSymbolThe functionfdu,dv in equation (28)IRDIFdu/dv−dv/duIRLlndu−lndvIRLUdu−dv/mindu,dvIRLFdu−dv/du×dvIRLA2du−dv/du+dvIRLD1ln1+du−dvIRGAlndu−dv/du×dvIRAdu−1/2−dv−1/22IRBdu1/2−dv1/22Most of the well-known degree-based irregularity indices can be obtained from the following general setting:(4)IRG=∑uv∈EGfdu,dv,where fdu,dv is an appropriately selected function.


## 2. Results for the Crystallographic Structure

In this section, we will compute the irregularity indices of some popular crystallographic structure of moleculesCu2Op,q,r.From [18], we know that the graph Cu2Op,q,r contains 6pqr+pq+qr+pr+q+p+r+1 vertices and 8pqr edges.We begin with obtaining formulas for irregularity measures ofIRRCu2Op,q,r, IRRtCu2Op,q,r, and IRFCu2Op,q,r.Theorem 1.
LetCu2Op,q,r be the crystallographic structure. Then, the irregularity indices of Cu2Op,q,r are(5)IRRCu2Op,q,r=16pqr−8qr−8pq−8pr+12p+12q+12r−16,(6)IRRtCu2Op,q,r=8pqr−4qr−4pq−4pr+6p+6q+6r−8,(7)IRFCu2Op,q,r=32pqr−16qr−16pq−16pr+20p+20q+20r−24.Proof.
Consider the graph crystallographic structureCu2Op,q,r. From the graph of Cu2Op,q,r crystallographic structure, we can see that there are three partitions, V1=v∈VCu2Op,q,r|dv=1, V2=v∈VCu2Op,q,r|dv=2, and V3=v∈VCu2Op,q,r|dv=4. The edge set of Cu2Op,q,r can be partitioned as follows:(8)E1=e=uv∈ECu2Op,q,r|du=1anddv=2,E2=e=uv∈ECu2Op,q,r|du=2anddv=2,E3=e=uv∈ECu2Op,q,r|du=2anddv=4.
From the molecular graph ofCu2Op,q,r, we can observe in Figures 1–5 that E1=4p+4q+4r−8,E2=4pq+4pr+4qr−8p−8q−8r+12, and E3=42pqr−pq−pr−qr+p+q+r−1. Thus, by the definition of Albertson index for Cu2Op,q,r, we have(9)IRRCu2Op,q,r=∑uv∈ECu2Op,q,rdu−dv=∑uv∈E1Cu2Op,q,rdu−dv+∑uv∈E2Cu2Op,q,rdu−dv+∑uv∈E3Cu2Op,q,rdu−dv=4p+4q+4r−8×1−2+42pqr−pq−pr−qr+p+q+r−1×2−4=16pqr−8qr−8pq−8pr+12p+12q+12r−16,which is the required formula result (5).
By the definition of total irregularity measure ofCu2Op,q,r, we have(10)IRRtCu2Op,q,r=12∑uv∈ECu2Op,q,rdu−dv=12∑uv∈ECu2Op,q,rdu−dv+12∑uv∈E2Cu2Op,q,rdu−dv+12∑uv∈E3Cu2Op,q,rdu−dv=12×4p+4q+4r−8×1−2+12×42pqr−pq−pr−qr+p+q+r−1×2−4=8pqr−4qr−4pq−4pr+6p+6q+6r−8,which is the required formula result (7).
By the definition of irregularity indexIRF for Cu2Op,q,r, we have(11)IRFCu2Op,q,r=∑uv∈ECu2Op,q,rdu−dv2=∑uv∈E1Cu2Op,q,rdu−dv2+∑uv∈E2Cu2Op,q,rdu−dv2+∑uv∈E3Cu2Op,q,rdu−dv2=4p+4q+4r−8×1−22+42pqr−pq−pr−qr+p+q+r−1×2−42=32pqr−16qr−16pq−16pr+20p+20q+20r−24,which is the required formula result (8).
Note that throughout the paper, colors red, blue, and green representIRR,IRRt, and IRF, respectively.
The next example represents some values of the calculated irregularity indices ofIRRCu2Op,q,r, RRtCu2Op,q,r, and IRFCu2Op,q,r for some test values of p=q=r=1.Example 1.
LetCu2O1,1,1 be crystallographic structure described in Figure 2. Then,(12)IRRCu2O1,1,1=12,IRRtCu2O1,1,1=6,IRFCu2O1,1,1=20.Theorem 2.
LetCu2Op,q,r be the crystallographic structure. Then, the irregularity indices of Cu2Op,q,r are(13)IRLCu2Op,q,r=5.5451774444792pqr−2.7725887222396qr−2.7725887222396pq−2.7725887222396pr+5.5451774444792p+5.5451774444792q+5.54517744447927−8.3177661667188.IRD1Cu2Op,q,r=8.7888983093448pqr−4.3944491546724pq−4.3944491546724pr−4.3944491546724qr+7.167037876912p+7.167037876912q+7.167037876912r−9.9396265991516.IRGACu2Op,q,r=1.8845285705024pqr−0.9422642852512pq−0.9422642852512pr−0.9422642852512qr+1.177830356564p+1.177830356564q+1.177830356564r−0.7066982139384.Proof.
By definition ofIRL irregularity index of Cu2Op,q,r, we have(14)IRLCu2Op,q,r=∑uv∈ECu2Op,q,rlndu−lndv=∑uv∈E1Cu2Op,q,rlndu−lndv+∑uv∈E2Cu2Op,q,rlndu−lndv+∑uv∈E3Cu2Op,q,rlndu−lndv=4p+4q+4r−8×ln1−ln2+42pqr−pq−pr−qr+p+q+r−1×ln2−ln4=5.5451774444792pqr−2.7725887222396qr−2.7725887222396pq−2.7725887222396pr+5.5451774444792p+5.5451774444792q+5.54517744447927r−8.3177661667188.
By applying the definition ofIRD1 for Cu2Op,q,r, we have(15)IRD1Cu2Op,q,r=∑uv∈ECu2Op,q,rln1+du−dv=∑uv∈E1Cu2Op,q,rln1+du−dv+∑uv∈E2Cu2Op,q,rln1+du−dv+∑uv∈E3Cu2Op,q,rln1+du−dv=4p+4q+4r−8×ln1+1−2+42pqr−pq−pr−qr+p+q+r−1×ln1+2−4=8.7888983093448pqr−4.3944491546724pq−4.3944491546724pr−4.3944491546724qr+7.167037876912p+7.167037876912q+7.167037876912r−9.9396265991516.
Finally, by applying the definition ofIRGA for Cu2Op,q,r, we have(16)IRGACu2Op,q,r=∑uv∈ECu2Op,q,rlndu+dv2du×dv=∑uv∈E1Cu2Op,q,rlndu+dv2du×dv+∑uv∈E2Cu2Op,q,rlndu+dv2du×dv+∑uv∈E3Cu2Op,q,rlndu+dv2du×dv=4p+4q+4r−8×ln1+221×2+4pq+4pr+4qr−8p−8q−8r+12×ln2+222×2+42pqr−pq−pr−qr+p+q+r−1××ln2+422×4=1.8845285705024pqr−0.9422642852512pq−0.9422642852512pr−0.9422642852512qr+1.177830356564p+1.177830356564q+1.177830356564r−0.7066982139384.
Note that throughout the paper, colors violet, yellow, and Peru representIRL,IRD1, and IRGA, respectively.Theorem 3.
LetCu2Op,q,r be the crystallographic structure. Then, the irregularity indices of Cu2Op,q,r are(17)IRACu2Op,q,r=4p+4q+4r−8×32−2+2pqr−pq−pr−qr+p+q+r−1×3−22,IRBCu2Op,q,r=4p+4q+4r−8×3−22+2pqr−pq−pr−qr+p+q+r−1×24−162,IRDIFCu2Op,q,r=12pqr−6qr−6pq−6pr+12p+12q+12r−18.Proof.
By applying the definition ofIRA for Cu2Op,q,r, we have(18)IRACu2Op,q,r=∑uv∈ECu2Op,q,r1du−1dv2=∑uv∈E1Cu2Op,q,r1du−1dv2+∑uv∈E2Cu2Op,q,r1du−1dv2+∑uv∈E3Cu2Op,q,r1du−1dv2=4p+4q+4r−8×1−122+42pqr−pq−pr−qr+p+q+r−1×12−142=4p+4q+4r−8×32−2+2pqr−pq−pr−qr+p+q+r−1×3−22.
By applying the definition ofIRB for Cu2Op,q,r, we have(19)IRBCu2Op,q,r=∑uv∈ECu2Op,q,rdu−dv2=∑uv∈E1Cu2Op,q,rdu−dv2+∑uv∈E2Cu2Op,q,rdu−dv2+∑uv∈E3Cu2Op,q,rdu−dv2=4p+4q+4r−8×1−22+42pqr−pq−pr−qr+p+q+r−1×2−42=4p+4q+4r−8×3−22+2pqr−pq−pr−qr+p+q+r−1×24−162.
Finally, by applying the definition ofIRDIF for Cu2Op,q,r, we have(20)IRDIFCu2Op,q,r=∑uv∈ECu2Op,q,rdudv−dvdu=∑uv∈E1Cu2Op,q,rdudv−dvdu+∑uv∈E2Cu2Op,q,rdudv−dvdu+∑uv∈E3Cu2Op,q,rdudv−dvdu=4p+4q+4r−8×12−21+42pqr−pq−pr−qr+p+q+r−1×24−42=12pqr−6qr−6pq−6pr+12p+12q+12r−18.
Note that throughout the paper, colors aquamarine, black, and blue violet representIRA,IRB, and IRDIF, respectively.Theorem 4.
LetCu2Op,q,r be the crystallographic structure. Then, the irregularity indices of Cu2Op,q,r are(21)IRLUCu2Op,q,r=8pqr−4qr−4pq−4pr+8p+8q+8r−12,IRLACu2Op,q,r=8pqr3−4qr3−4pq3−4pr3+8p3+8q3+8r3−4,IRLFCu2Op,q,r=4p+4q+4r−8×2+2pqr−pq−pr−qr+p+q+r−1×22.Proof.
By definition ofIRLU irregularity index of Cu2Op,q,r, we have(22)IRLUCu2Op,q,r=∑uv∈ECu2Op,q,rdu−dvmindu,dv=∑uv∈E1Cu2Op,q,rdu−dvmindu,dv+∑uv∈E2Cu2Op,q,rdu−dvmindu,dv+∑uv∈E3Cu2Op,q,rdu−dvmindu,dv=4p+4q+4r−8×1−2min1,2+42pqr−pq−pr−qr+p+q+r−1×2−4min2,4=8pqr−4qr−4pq−4pr+8p+8q+8r−12.
By applying the definition ofIRLA for Cu2Op,q,r, we have(23)IRLACu2Op,q,r=∑uv∈ECu2Op,q,r2du−dvdu+dv=∑uv∈E1Cu2Op,q,r2du−dvdu+dv+∑uv∈E2Cu2Op,q,r2du−dvdu+dv+∑uv∈E3Cu2Op,q,r2du−dvdu+dv=4p+4q+4r−8×1−21+2+42pqr−pq−pr−qr+p+q+r−1×2−42+4=8pqr3−4qr3−4pq3−4pr3+8p3+8q3+8r3−4.
By applying the definition ofIRLF for Cu2Op,q,r, we have(24)IRLFCu2Op,q,r=∑uv∈ECu2Op,q,rdu−dvdu×dv=∑uv∈E1Cu2Op,q,rdu−dvdu×dv+∑uv∈E2Cu2Op,q,rdu−dvdu×dv+∑uv∈E3Cu2Op,q,rdu−dvdu×dv=4p+4q+4r−8×1−21×2+42pqr−pq−pr−qr+p+q+r−1×2−42×4=4p+4q+4r−8×2+2pqr−pq−pr−qr+p+q+r−1×22.
Note that throughout the paper, colors dark orange, fuchsia, and gray representIRLU,IRLA, and IRLF, respectively.
Next example represents some values of the calculated irregularity indices ofIRLUCu2Op,q,r, IRLACu2Op,q,r,IRLFCu2Op,q,r, and IRDIFCu2Op,q,r for some test values of p=3,q=2, and r=3.Example 2.
LetCu2O3,2,3 be crystallographic structure described in Figure 3.
Then,(25)IRLUCu2O3,2,3=112,IRLACu2O3,2,3=37.3333333333333,IRLFCu2O3,2,3=79.1959594928933,IRDIFCu2O3,2,3=168.


## 3. Results for the Crystallographic Structure of Titanium Difluoride

In this section, we compute irregularity measures of crystal structure of titanium difluoride ofTiF2p,q,r.Titanium difluoride is a water-insoluble titanium hotspot for use in oxygen-delicate applications, for example, metal generation. The concoction chart of the crystallographic structure ofTiF2p,q,r is depicted in Figure 1 graph unit cell of Cu2O (see [29]). In Figures 4 and 5, red dots are for F atoms and green dots are for Ti atoms.From [18], we know that the graph TiF2p,q,r contains 12pqr+2pq+2pr+2qr+p+q+r+1 vertices and 32pqr edges.Here, we obtain some formulas for irregularity measures ofIRRTiF2p,q,r, IRRtTiF2p,q,r, and IRFTiF2p,q,r.Theorem 5.
LetTiF2p,q,r be the crystallographic structure of titanium difluoride. Then, the irregularity indices of TiF2p,q,r are(26)IRRTiF2p,q,r=128pqr−64qr−64pq−64pr+48p+48q+48r−56,(27)IRRtTiF2p,q,r=64pqr−32qr−32pq−32pr+24p+24q+24r−28,(28)IRFTiF2p,q,r=512pqr−256qr−256pq−256pr+160p+160q+160r−152.Proof.
Consider the graph crystallographic structure of titanium difluorideTiF2p,q,r. From the graph of TiF2p,q,r crystallographic structure of titanium difluoride, we can see that there are four partitions, V1=v∈VTiF2p,q,r|dv=1, V2=v∈VTiF2p,q,r|dv=2, V3=v∈VTiF2p,q,r|dv=4, and V4=v∈VTiF2p,q,r|dv=8. The edge set of TiF2p,q,r can be partitioned as follows:(29)E1=e=uv∈ETiF2p,q,r|du=1anddv=4,E2=e=uv∈ETiF2p,q,r|du=2anddv=4,E3=e=uv∈ETiF2p,q,r|du=4anddv=4,E4=e=uv∈ETiF2p,q,r|du=4anddv=8.
From the molecular graph ofTiF2p,q,r, we can observe in Figures 4 and 5 that E1=8,E2=8p+q+r−3, E3=16pq+pr+qr−16p+q+r+24, and E2=32pqr−16pq+pr+qr+8p+q+r−8. By applying the definition of Albertson index for TiF2p,q,r, we have(30)IRRTiF2p,q,r=∑uv∈ETiF2p,q,rdu−dv=∑uv∈E1TiF2p,q,rdu−dv+∑uv∈E2TiF2p,q,rdu−dv+∑uv∈E3TiF2p,q,rdu−dv+∑uv∈E4TiF2p,q,rdu−dv=8×1−4+8p+q+r−3×2−4+32pqr−16pq+pr+qr+8p+q+r−8×|4−8|=128pqr−64qr−64pq−64pr+48p+48q+48r−56,which is the required result (26). By applying the definition of total irregularity measure for TiF2p,q,r, we have(31)IRRtCu2Op,q,r=12∑uv∈ETiF2p,q,rdu−dv=12∑uv∈E1TiF2p,q,rdu−dv+12∑uv∈E2TiF2p,q,rdu−dv+12∑uv∈E3TiF2p,q,rdu−dv+12∑uv∈E4TiF2p,q,rdu−dv=12×8×1−4+12×8p+q+r−3×2−4+32pqr−16pq+pr+qr+8p+q+r−8×12×4−8=64pqr−32qr−32pq−32pr+24p+24q+24r−28,which is the required result (27).
By applying the definition ofIRF for TiF2p,q,r, we have(32)IRFTiF2p,q,r=∑uv∈ETiF2p,q,rdu−dv2=∑uv∈E1TiF2p,q,rdu−dv2+∑uv∈E2CTiF2p,q,rdu−dv2+∑uv∈E3TiF2p,q,rdu−dv2+∑uv∈E4TiF2p,q,rdu−dv2=8×1−42+8p+q+r−3×2−42+32pqr−16pq+pr+qr+8p+q+r−8×4−82=512pqr−256qr−256pq−256pr+160p+160q+160r−152,which is the required result (28).
The next example represents some values of the calculated irregularity indices ofIRALTiF2p,q,r, IRRtTiF2p,q,r, and IRFTiF2p,q,r for some test values of p=q=r=1.Example 3.
LetTiF21,1,1 be the crystallographic structure of titanium difluoride described in Figure 4. Then,(33)IRALTiF21,1,1=24,IRRtTiF21,1,1=12,IRFTiF21,1,1=72.Theorem 6.
LetTiF2p,q,r be the crystallographic structure of titanium difluoride. Then, the irregularity indices of TiF2p,q,r are(34)IRLTiF2p,q,r=22.1807097779168pqr−11.0903548889584qr−11.0903548889584pq−11.0903548889584pr+11.0903548889584p+11.0903548889584q+11.0903548889584r−11.0903548889576,IRD1TiF2p,q,r=51.5020131978912pqr−25.7510065989456pq−25.7510065989456pr−25.7510065989456qr+21.6644016088176p+21.6644016088176q+21.6644016088176r−28.151843338548,IRGATiF2p,q,r=1.8845285705024pqr−0.9422642852512pq−0.9422642852512pr−0.9422642852512qr+1.177830356564p+1.177830356564q+1.177830356564r−0.0993801599888.Proof.
By applying the definition ofIRL for TiF2p,q,r, we have(35)IRLTiF2p,q,r=∑uv∈ETiF2p,q,rlndu−lndv=∑uv∈E1TiF2p,q,rlndu−lndv+∑uv∈E2TiF2p,q,rlndu−lndv+∑uv∈E3TiF2p,q,rlndu−lndv=8×ln1−ln4+8p+q+r−3×ln2−ln4+32pqr−16pq+pr+qr+8p+q+r−8×|ln4−ln8|=22.1807097779168pqr−11.0903548889584qr−11.0903548889584pq−11.0903548889584pr+11.0903548889584p+11.0903548889584q+11.0903548889584r−11.0903548889576.
By applying the definition ofIRD1 for TiF2p,q,r, we have(36)IRD1TiF2p,q,r=∑uv∈ETiF2p,q,rln1+du−dv=∑uv∈E1TiF2p,q,rln1+du−dv+∑uv∈E2TiF2p,q,rln1+du−dv+∑uv∈E3TiF2p,q,rln1+du−dv+∑uv∈E3TiF2p,q,rln1+du−dv=8×ln1+1−4+8p+q+r−3×ln1+2−4+32pqr−16pq+pr+qr+8p+q+r−8×ln1+4−8=51.5020131978912pqr−25.7510065989456pq−25.7510065989456pr−25.7510065989456qr+21.6644016088176p+21.6644016088176q+21.6644016088176r−28.151843338548.
Finally, by applying the definition ofIRGA for TiF2p,q,r, we have(37)IRGATiF2p,q,r=∑uv∈ETiF2p,q,rlndu+dv2du×dv=∑uv∈E1TiF2p,q,rlndu+dv2du×dv+∑uv∈E2TiF2p,q,rlndu+dv2du×dv+∑uv∈E3TiF2p,q,rlndu+dv2du×dv+∑uv∈E4TiF2p,q,rlndu+dv2du×dv=8×ln1+421×4+8p+q+r−3×ln2+422×4+16pq+16pr+16qr−16p−16q−16r+24××ln4+424×4+32pqr−16pq+pr+qr+8p+q+r−8×ln4+824×8=1.8845285705024pqr−0.9422642852512pq−0.9422642852512pr−0.9422642852512qr+1.177830356564p+1.177830356564q+1.177830356564r−0.0993801599888.Theorem 7.
LetTiF2p,q,r be the crystallographic structure of titanium difluoride. Then, the irregularity indices of TiF2p,q,r are(38)IRATiF2p,q,r=4p+4q+4r−8×6−42+2pqr−pq−pr−qr+p+q+r−1×−24+38+2,IRBTiF2p,q,r=4p+4q+4r−8×48−322+2pqr−pq−pr−qr+p+q+r−1×12−82+8,IRDIFTiF2p,q,r=48pqr−24qr−24pq−24pr+24p+24q+24r−18.Proof.
By applying the definition ofIRA for TiF2p,q,r, we have(39)IRATiF2p,q,r=∑uv∈ETiF2p,q,r1du−1dv2=∑uv∈E1TiF2p,q,r1du−1dv2+∑uv∈E2TiF2p,q,r1du−1dv2+∑uv∈E3TiF2p,q,r1du−1dv2+∑uv∈E4TiF2p,q,r1du−1dv2=8×1−142+8p+q+r−3×12−142+32pqr−16pq+pr+p+qqr+8+r−8×14−182=4p+4q+4r−8×6−42+2pqr−pq−pr−qr+p+q+r−1×−24+38+2.
By applying the definition ofIRB for TiF2p,q,r, we have(40)IRBTiF2p,q,r=∑uv∈ETiF2p,q,rdu−dv2=∑uv∈E1TiF2p,q,rdu−dv2+∑uv∈E2TiF2p,q,rdu−dv2+∑uv∈E3TiF2p,q,rdu−dv2+∑uv∈E3TiF2p,q,rdu−dv2=8×1−42+8p+q+r−3×2−42+32pqr−16pq+pr+qr+8p+q+r−8×4−82=4p+4q+4r−8×48−322+2pqr−pq−pr−qr+p+q+r−1×12−82+8.
By applying the definition ofIRDIF for TiF2p,q,r, we have(41)IRDIFTiF2p,q,r=∑uv∈ETiF2p,q,rdudv−dvdu=∑uv∈E1TiF2p,q,rdudv−dvdu+∑uv∈E2TiF2p,q,rdudv−dvdu+∑uv∈E3TiF2p,q,rdudv−dvdu+∑uv∈E4TiF2p,q,rdudv−dvdu=8×14−41+8p+q+r−3×24−42+32pqr−16pq+pr+p+qqr+8+r−8×48−84=48pqr−24qr−24pq−24pr+24p+24q+24r−18.Theorem 8.
LetTiF2p,q,r be the crystallographic structure of titanium difluoride. Then, the irregularity indices of TiF2p,q,r are(42)IRLUTiF2p,q,r=32pqr−16qr−16pq−16pr+16p+16q+16r−8,(43)IRLATiF2p,q,r=32pqr3−16qr3−16pq3−16pr3+16p3+16q3+16r3−8,(44)IRLFTiF2p,q,r=p+q+r−3×42+16pqr−8pq−8pr−8qr+4p+4q+4r−4×2+42.Proof.
By applying the definition ofIRLU for TiF2p,q,r, we have(45)IRLUTiF2p,q,r=∑uv∈ETiF2p,q,rdu−dvmindu,dv=∑uv∈E1TiF2p,q,rdu−dvmindu,dv+∑uv∈E2TiF2p,q,rdu−dvmindu,dv+∑uv∈E3TiF2p,q,rdu−dvmindu,dv+∑uv∈E4TiF2p,q,rdu−dvmindu,dv=8×1−4min1,4+8p+q+r−3×2−4min2,4+32pqr−16pq+pr+p+qqr+8+r−8×4−8min4,8=32pqr−16qr−16pq−16pr+16p+16q+16r−8,which is the required result (42).By applying the definition of IRLA for TiF2p,q,r, we have(46)IRLATiF2p,q,r=∑uv∈ETiF2p,q,r2du−dvdu+dv=∑uv∈E1TiF2p,q,r2du−dvdu+dv+∑uv∈E2TiF2p,q,r2du−dvdu+dv+∑uv∈E3TiF2p,q,r2du−dvdu+dv+∑uv∈E4TiF2p,q,r2du−dvdu+dv=8×1−21+2+8p+q+r−3×2−42+4+32pqr−16pq+pr+p+qqr+8+r−8×4−84+8=32pqr3−16qr3−16pq3−16pr3+16p3+16q3+16r3−8,which is the required result (43).
By applying the definition ofIRLF for TiF2p,q,r, we have(47)IRLFTiF2p,q,r=∑uv∈ETiF2p,q,rdu−dvdu×dv=∑uv∈E1TiF2p,q,rdu−dvdu×dv+∑uv∈E2TiF2p,q,rdu−dvdu×dv+∑uv∈E3TiF2p,q,rdu−dvdu×dv+∑uv∈E4TiF2p,q,rdu−dvdu×dv=8×1−21×2+8p+q+r−3××2−42×4+32pqr−16pq+pr+p+qqr+8+r−8×4−84×8=p+q+r−3×42+16pqr−8pq−8pr−8qr+4p+4q+4r−4×2+42,which is the required formula result (44).
Finally, the next example represents some values of the calculated irregularity indices ofIRLUTiF2p,q,r, IRLATiF2p,q,r, IRLFTiF2p,q,r, and IRDIFTiF2p,q,r for some test values of p=4,q=1, and r=2.Example 4.
LetTiF24,1,2 be the crystallographic structure of titanium difluoride in Figure 5. Then,(48)IRLUTiF24,1,2=136,IRLATiF24,1,2=40,IRLFTiF24,1,2=84.8528137423857,IRDIFTiF24,1,2=294.


## 4. Comparisons and Discussion

The main motivation comes from the fact that graphs of the irregularity indices show close accurate results about properties like entropy, standard enthalpy, vaporization, and acentric factors of octane isomers [8]. Irregularity indices may help to measure the chemical, biological, and nano properties which are widely popular in developing areas.Through the means of a graph structural analysis and derivation, we compute some irregularity measures of crystallographic structure of moleculesCu2Op,q,r and the crystallographic structure of titanium difluoride of TiF2p,q,r. Similar works have been done in [12, 13, 17]. From Tables 2–9, we can easily see that all indices are in increasing order as the values of p,q, and r. On the other hand, indices showed higher values for Cu2Op,q,r and TiF2p,q,r. The graphical representations of irregularity indices of Cu2Op,q,r and TiF2p,q,r are depicted in Figures 6–13 for certain values of p,q, and r. Moreover, we presented the comparison of all irregularity indices in Figures 14 and 15 for Cu2Op,q,r and TiF2p,q,r, respectively.Table 2
Comparison ofIRR,IRRt, and IRF for crystallographic structure Cu2Op,q,r.
p,q,rIRRIRRtIRF1,1,1126202,2,288441603,3,33081545884,4,47683841496Table 3
Comparison ofIRL,IRD1, and IRGA for crystallographic structure Cu2Op,q,r.
p,q,rIRLIRD1IRGA1,1,15.54517744447927.1670378769120.47113214262552,2,236.043653389114850.640397281013.0623589270663,3,3116.4487263340632173.21384146921129.89377499513624,4,4280.0314609461996427.620760297584423.7921732025895Table 4
Comparison ofIRA,IRB, and IRDIF for crystallographic structure Cu2Op,q,r.
p,q,rIRAIRBIRDIF1,1,10.51471862576142.0588745030457122,2,22.916738879314815.0984130223353783,3,310.465945390482452.84444557817342524,4,417.6720061511424131.7679681949257606Table 5
Comparison ofIRLU,IRLA, and IRLF for crystallographic structure Cu2Op,q,r.
p,q,rIRLUIRLAIRLF1,1,182.058874503045716.97056274847712,2,25217.3333333333333124.45079348883243,3,316856435.57777721091334,4,440434.66666666666671086.1160159025371Table 6
Comparison ofIRR,IRRt, and IRF for crystallographic structure of titanium difluoride TiF2p,q,r.
p,q,rIRRIRRtIRF1,1,12412722,2,248824418323,3,32104105282004,4,45640282022248Table 7
Comparison ofIRL,IRD1, and IRGA for the crystallographic structure of titanium difluoride TiF2p,q,r.
p,q,rIRLIRD1IRGA2,2,283.1776616671888165.834566314357869.5542055515183,3,3363.2091226133884598.8277711408108440.38099840727114,4,4975.951230228341511.82097596726381291.2077912630243Table 8
Comparison ofIRA,IRB, and IRDIF for the crystallographic structure of titanium difluoride TiF2p,q,r.
p,q,rIRAIRBIRDIF1,1,13.394029611437219.6669555172591122,2,27.683351492782458.0992795741124783,3,312.3587123434488108.88485064924022524,4,417.6774714763168180.259166754825Table 9
Comparison ofIRLU,IRLA, and IRLF for the crystallographic structure of titanium difluoride TiF2p,q,r.
p,q,rIRLUIRLAIRLF1,1,1242.666666666666702,2,215245.333333333333390.50966799187813,3,356818484.66608896548194,4,41464482.66666666666671018.2337649086285Figure 6
Comparison ofIRR,IRRt, and IRF of crystallographic structure Cu2Op,q,r.
(a)(b)(c)Figure 7
Comparison ofIRL,IRD1, and IRGA for crystallographic structure Cu2Op,q,r.
(a)(b)(c)Figure 8
Comparison ofIRA,IRB, and IRDIF for crystallographic structure Cu2Op,q,r.
(a)(b)(c)Figure 9
Comparison ofIRLU,IRLA, and IRLF for crystallographic structure Cu2Op,q,r.
(a)(b)(c)Figure 10
Comparison ofIRR,IRRt, and IRF for crystallographic structure of titanium difluoride TiF2p,q,r.
(a)(b)(c)Figure 11
Comparison ofIRL,IRD1, and IRGA for crystallographic structure of titanium difluoride TiF2p,q,r.
(a)(b)(c)Figure 12
Comparison ofIRA,IRB, and IRDIF for crystallographic structure of titanium difluoride TiF2p,q,r.
(a)(b)(c)Figure 13
Comparison ofIRLU,IRLA, and IRLF for crystallographic structure of titanium difluoride TiF2p,q,r.
(a)(b)(c)Figure 14
The comparison of all irregularity indices for crystallographic structureCu2Op,q,r.
(a)(b)(c)Figure 15
The comparison of all irregularity indices for crystallographic structure of titanium difluorideTiF2p,q,r.
(a)(b)(c)


## 5. Conclusion

In this paper, we studied popular crystallographic structure of molecules and also applied analytical methods to compute the irregularity measures for crystallographic structure of moleculesCu2Op,q,r and crystallographic structure of titanium difluoride of TiF2p,q,r.



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