# Irregularity Measures for Metal-Organic Networks

Authors: Xuan Guo; Yu-Ming Chu; Muhammad Khalid Hashmi; Abaid Ur Rehman Virk; Jingjng Li
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Topological index plays an important role in predicting physicochemical properties of a molecular structure. With the help of the topological index, we can associate a single number with a molecular graph. Drugs and other chemical compounds are frequently demonstrated as different polygonal shapes, trees, graphs, etc. In this paper, we will compute irregularity indices for metal-organic networks.


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## Body

## 1. Introduction

Every planet is a blend of various kinds of parts, and every part has a significant commitment in the composition of the earth. The most significant parts in the earth are hydrogen, oxygen, and nitrogen. Subatomic hydrogen is one of the segments that has cordial situation and is a cutting edge wellspring of vitality [1, 2]. As a gas, it can likewise be used in energy units to control engines. Among the various kinds of gases, hydrogen needs smell that makes any hole acknowledgment practically difficult to human creatures. The ongoing standards settled by the United States Vitality Sector put emphasis on the speed proficient of the gadget that can detect one percent by volume of drab and also scent-free atomic hydrogen in climate in less than sixty seconds in particular [3–6].Jang et al. [7] presented an exceptionally quick hydrogen recognizing device comprising organic ligands and metals recognized with the assistance of palladium nanowires which can perceive hydrogen stages lower than 1 percent in just seven seconds. Moreover, other than distinguishing and detecting, the MONs appear exceptionally valuable with synthetic and physical properties, for example, grafting active groups [8], changing natural ligands [9], impregnating reasonable active materials [10], postsynthetic ligands, ion trade [11], and getting ready composites with useable substance [12].Graph theory provides the interesting appliance in mathematical chemistry that is used to compute the various kinds of chemical compounds by means of graph theory and predict their various properties [13–19]. One of the most important tools in the chemical graph theory is a topological index, which is useful in predicting the chemical and physical properties of the underlying chemical compound, such as boiling point, strain energy, rigidity, heat of evaporation, and tension [20, 21]. A graph having no loop or multiple edge is known as a simple graph. A molecular graph is a simple graph in which atoms and bounds are represented by the vertex and edge sets, respectively. The degree of the vertex is the number of edges attached with that vertex. These properties of various objects is of primary interest. Winner, in 1947, introduced the concept of the first topological index while finding the boiling point. In 1975, Gutman gave a remarkable identity [22] about Zagreb indices. Hence, these two indices are among the oldest degree-based descriptors, and their properties are extensively investigated. The mathematical formulae of these indices are(1)M1G=∑uv∈EGdu+dv,M2G=∑uv∈EGdu×dv.A topological index is known as the irregularity index [23] if the value of the topological index of the graph is greater than or equal to zero, and the topological index of the graph is equal to zero if and only if the graph is regular. The irregularity indices are given in Table 1. Most of the irregularity indices are from the family of degree-based topological indices and are used in quantitative structure activity relationship modeling.Table 1
Definitions of irregularity indices.
Irregularity indexMathematical formVAR∑uϵVdu−2m/n2=M1G/n−2m/n2AL∑uv∈EGdu−dvIR1∑u∈Vdu3−2m/n∑u∈Vdu2=FG−2m/nM1GIR2∑uv∈EGdudv/m−2m/n=M2G/m−2m/nIRF∑uv∈EGdu−dv2=FG−2M2GIRFWIRFG/M2GIRA∑uv∈EGdu−1/2−dv−1/22=n−2RGIRB∑uv∈EGdu1/2−dv1/22=M1G−2RRGIRC∑uv∈EGdudv/m−2m/n=RRG/m−2m/nIRDIF∑uv∈EGdu/dv−dv/du=∑i<jmi,jj/i−i/jIRL∑uv∈EGlndu−lndv=∑i<jmi,jlnj/iIRLU∑uv∈EGdu−dv/mindu,dv=∑i<jmi,jlnj−i/iIRLF∑uv∈EGdu−dv/dudv=∑i<jmi,jj−i/ijIRLA2∑uv∈EGdu−dv/du+dv=2∑i<jmi,jj−i/i+jIRDI∑uv∈EGln1+du−dv=∑i<jmi,jlni+j−1IRGA∑uv∈EGlndu+dv/2dudv∑i<jmi,ji+j/2ijFor more about topological indices, one can read [24–31].


## 2. Irregularity Indices for Metal-Organic Networks

In this section, we discuss metal-organic networks by means of a graph. The unit cell of the metal-organic network is given in Figure1. We give computational results of irregularity indices for two types of metal-organic networks MON1p and MON2p in the following two sections.Figure 1
Basic metal-organic network.


### 2.1. Irregularity Indices for Metal-Organic NetworkMON1p

This section is about irregularity indices of the metal-organic networkMON1p. The molecular graph of the metal-organic network MON1p for p=2 is given in Figure 2. We can observe from Figure 2 that there are four types of vertices present in the molecular graph of MON1p, i.e., 2, 3, 4, and 6. The cardinality of vertices 2, 3, 4, and 6 is 30p, 12, 12p−6, and 6p−6, respectively. The cardinality of the vertex set of MON1p is 48p, i.e., VMON1p=48p. There are four different types of edges present in the molecular graph of MON1p, i.e., 2,3, 2,4, 2,6, and 4,6. Their cardinalities are 36, 36p−12, 24p−24, and 12p−12, respectively. The cardinality of the edge set of MON1p is 72p−12, i.e., EMON1p=72p−12.Figure 2
MON1p for p=2.The edge partition of the metal-organic networkMON1p is given in Table 2.Table 2
Partition ofEMON1p.
(du,dv)Frequency(2, 3)36(2, 4)12 (3p − 1)(2, 6)24 (p − 1)(4, 6)12 (p − 1)Theorem 1.
LetG be the metal-organic network EMON1p. The irregularity indices are(1)
VARG=8p2−5p−1/4p2(2)
ALG=192p−108(3)
IR1G=6120p2−80p−17/p(4)
IR2G=864p−456/48p−72p−12/24p(5)
IRFG=576p−444(6)
IRFWG=48p−37/236p−19(7)
IRAG=48p−26+92+43p−106+62+83(8)
IRBG=528p−204−2246+722+483p−246+482+963(9)
IRCG=46p2+26p+122p2+83p2−42p−83p−36p2+12p−1/2p6p−1(10)
IRDIFG=127.9980p−61.9992(11)
IRLG=56.1840p−24.9516(12)
IRLUG=90p−48(13)
IRLFG=58.0668p−26.4012(14)
IRLAG=52.7976p−22.3992(15)
IRD1G=131.0064p−42.4776(16)
IRGAG=8.0172p−5.8716Proof.
Using definitions of irregularity indices given in Table1 and edge partition given in Table 2, we have(2)VARG=∑uϵVdu−2mn2=M1Gn−2mn2=528p−20448p−272p−1248p2=8p2−5p−14p2,ALG=∑uvϵEGdu−dv=2−336+2−436p−12+2−624p−1+4−612p−1=192p−108,IR1G=∑uϵVdu3−2mn∑uϵVdu2=FG−2mnM1G=2304p−1356−272p−1248p528p−204=6120p2−80p−17p,IR2G=∑uvϵEGdudvm−2mn=M2Gm−2mn=864p−45672p−12−272p−1248p=864p−45648p−72p−1224p,IRFG=∑uvϵEGdu−dv2=2−3236+2−4236p−12+2−6224p−1+4−6212p−1=576p−444,IRFWG=IRFGM2G=48p−37236p−19,IRAG=∑uvϵEGdu−1/2−dv−1/22=n−2RG=48p−292+3+6p−32+43−56=48p−26+92+43p−106+62+83,IRBG=∑uvϵEGdu1/2−dv1/22=M1DnPn−2RRG=528p−204−2722+483+246p−242+483−126=528p−204−2246+722+483p−246+482+963,IRCG=∑uvϵEGdudvm−2mn=RRGm−2mn=722+483+246p−242+483−12672p−12272p−1248p=46p2+26p+122p2+83p2−42p−83p−36p2+12p−12p6p−1,IRDIFG=∑uvϵEGdudv−dvdu=23−3236+24−4236p−12+26−6224p−1+46−6412p−1=127.9980p−61.9992,IRLG=∑uvϵEGlndu−lndv=ln2−ln336+ln2−ln436p−12+ln2−ln624p−1+ln4−ln612p−1=56.1840p−24.9516,IRLUG=∑uvϵEGdu−dvmindu,dv=2−3236+2−4236p−12+2−6224p−1+4−6412p−190p−48,IRLFG=∑uvϵEGdu−dvdu.dv=2−3636+2−4836p−12+2−61224p−1+4−62412p−1=58.0668p−26.4012,IRLAG=∑uvϵEG2du−dvdu+dv=22−3536+22−4636p−12+22−6824p−1+24−61012p−1=52.7976p−22.3992,IRD1G=∑uvϵEGln1+du−dv=ln1+2−336+ln1+2−436p−12+ln1+2−624p−1+ln1+4−612p−1=131.0064p−42.4776,IRGAG=∑uvϵEGlndu+dv2dudv=ln2+322×336+ln2+422×436p−1+ln2+622×624p−1+ln4+624×612p−1=8.0172p−5.8716.


### 2.2. Irregularity Indices for Metal-Organic NetworkMON2p

In this section, we will compute irregularity indices for the metal-organic networkMON2p. The molecular graph of the metal-organic network MON2p for p=2 is given in Figure 3. It is clear from Figure 3 that there are three types of vertices in the molecular graph of MON2p, i.e., 2, 3, and 4. The cardinality of vertices 2, 3, and 4 is 12p+18, 24p−12, and 12p−6, respectively. The cardinality of the vertex set of MON2p is 48p, i.e., VMON2p=48p. There are five different types of edges present in the molecular graph of MON2p, i.e., 2,3, 2,4, 3,3, 3,4, and 4,4. Their cardinalities are 12p+24, 12p+12, 24p−24, 12p−12, and 12p−12, respectively. The cardinality of the edge set of MON2p is 72p−12, i.e., EMON1p=72p−12.Figure 3
MON2p for p=2.The edge partition of the metal-organic networkMON2p is given in Table 3.Table 3
Partition ofEMON2p.
(du,dv)Frequency(2, 3)12 (p + 2)(2, 4)12 (p + 1)(3, 3)24 (p − 1)(3, 4)12 (p − 1)(4, 4)12 (p − 1)Theorem 2.
EMON2pLet G be the metal-organic network MON2p. The irregularity indices are(1)
VARG=2p2+p−1/4p2(2)
ALG=48p+36(3)
IR1G=624p2+10p−11/p(4)
IR2G=720p−312/48p−72p−12/24p(5)
IRFG=72p+60(6)
IRFWG=72p+60/720p−312(7)
IRAG=32p+16−226+43+32p−86−62+83(8)
IRBG=216p+108−2126+242+243p−486−482+483(9)
IRCG=26p2+46p+43p2+42p2−43p+42p−16p2−8p−1/2p6p−1(10)
IRDIFG=34.9992p+30.9996(11)
IRLG=16.8660p+14.3664(12)
IRLUG=21.9996p+20.0004(13)
IRLFG=16.8468p+14.8188(14)
IRLAG=16.2276p+14.1708(15)
IRD1G=29.8176p+21.5004(16)
IRGAG=1.0740p+1.0716Proof.
Using definitions of irregularity indices given in Table1 and edge partition given in Table 3, we have(3)VARG=∑uεVdu−2mn2=M1Gn−2mn2=456p−13248p−272p−1248p2=2p2+p−14p2,ALG=∑uvϵEGdu−dv=2−312p+24+2−412p+12+3−324p−24+3−412p−12+4−412p−12=48p+36,IR1G=∑uϵVdu3−2mn∑uϵVdu2=FG−2mnM1G=1512p−564−272p−1248p456p−132=624p2+10p−11p,IR2G=∑uvϵEGdudvm−2mn=M2Gm−2mn=720p−31272p−12−272p−1248p=720p−31248p−72p−1224p,IRFG=∑uvϵEGdu−dv2=2−3212p+24+2−4212p+12+3−3224p−24+3−4212p−12+4−4212p−12=72p+60,IRFWG=IRFGM2G=72p+60720p−312,IRAG=∑uvϵEGdu−1/2−dv−1/22=n−2RG=48p−28p−8+32+43+26p+32−43+46=32p+16−226+43+32p−86−62+83,IRBG=∑uvɛEGdu1/2−dv1/22=M1G−2RRG=456p−132−2120p−120+242+243+126p+246−243+242=216p+108−2126+242+243p−486−482+483,IRCG=∑uvϵEGdudvm−2mn=RRGm−2mn=120p−120+242+243+126p+246−243+24272p−12272p−1248p=26p2+46p+43p2+42p2−43p+42p−16p2−8p−12p6p−1,IRDIFG=∑uvϵEGdudv−dvdu=23−3212p+24+24−4212p+12+33−3324p−24+34−4312p−12+44−4412p−12,=34.9992p+30.9996,IRLG=∑uvϵEGlndu−lndv=ln2−ln312p+24+ln2−ln412p+12+ln3−ln324p−24+ln3−ln412p−12+ln4−ln412p−12=16.8660p+14.3664,IRLUG=∑uvϵEGdu−dvmindu,dv=2−3212p+24+2−4212p+24+3−3324p−24+3−4312p−12+4−4412p−12=21.9996p+20.0004,IRLFG=∑uvϵEGdu−dvdu⋅dv=2−3612p+24+2−4812p+12+3−3924p−24+3−41212p−12+4−41612p−12=16.8468p+14.8188,IRLAG=∑uvϵEG2du−dvdu+dv=22−3512p+24+22−4612p+12+23−3624p−24+23−4712p−12+24−4812p−12=16.2276p+14.1708,IRD1G=∑uvϵEGln1+du−dv=ln1+2−312p+24+ln1+2−412p+12+ln1+3−324p−24+ln1+|3−412p−12+ln1+4−412p−12=29.8176p+21.5004,IRGAG=∑uvϵEGlndu+dv2dudv=ln2+322×312p+24+ln2+422×412p+12+ln3+323×324p−24+ln3+423×412p−12+ln4+424×412p−12=1.0740p+1.0716.


## 2.1. Irregularity Indices for Metal-Organic NetworkMON1p

This section is about irregularity indices of the metal-organic networkMON1p. The molecular graph of the metal-organic network MON1p for p=2 is given in Figure 2. We can observe from Figure 2 that there are four types of vertices present in the molecular graph of MON1p, i.e., 2, 3, 4, and 6. The cardinality of vertices 2, 3, 4, and 6 is 30p, 12, 12p−6, and 6p−6, respectively. The cardinality of the vertex set of MON1p is 48p, i.e., VMON1p=48p. There are four different types of edges present in the molecular graph of MON1p, i.e., 2,3, 2,4, 2,6, and 4,6. Their cardinalities are 36, 36p−12, 24p−24, and 12p−12, respectively. The cardinality of the edge set of MON1p is 72p−12, i.e., EMON1p=72p−12.Figure 2
MON1p for p=2.The edge partition of the metal-organic networkMON1p is given in Table 2.Table 2
Partition ofEMON1p.
(du,dv)Frequency(2, 3)36(2, 4)12 (3p − 1)(2, 6)24 (p − 1)(4, 6)12 (p − 1)Theorem 1.
LetG be the metal-organic network EMON1p. The irregularity indices are(1)
VARG=8p2−5p−1/4p2(2)
ALG=192p−108(3)
IR1G=6120p2−80p−17/p(4)
IR2G=864p−456/48p−72p−12/24p(5)
IRFG=576p−444(6)
IRFWG=48p−37/236p−19(7)
IRAG=48p−26+92+43p−106+62+83(8)
IRBG=528p−204−2246+722+483p−246+482+963(9)
IRCG=46p2+26p+122p2+83p2−42p−83p−36p2+12p−1/2p6p−1(10)
IRDIFG=127.9980p−61.9992(11)
IRLG=56.1840p−24.9516(12)
IRLUG=90p−48(13)
IRLFG=58.0668p−26.4012(14)
IRLAG=52.7976p−22.3992(15)
IRD1G=131.0064p−42.4776(16)
IRGAG=8.0172p−5.8716Proof.
Using definitions of irregularity indices given in Table1 and edge partition given in Table 2, we have(2)VARG=∑uϵVdu−2mn2=M1Gn−2mn2=528p−20448p−272p−1248p2=8p2−5p−14p2,ALG=∑uvϵEGdu−dv=2−336+2−436p−12+2−624p−1+4−612p−1=192p−108,IR1G=∑uϵVdu3−2mn∑uϵVdu2=FG−2mnM1G=2304p−1356−272p−1248p528p−204=6120p2−80p−17p,IR2G=∑uvϵEGdudvm−2mn=M2Gm−2mn=864p−45672p−12−272p−1248p=864p−45648p−72p−1224p,IRFG=∑uvϵEGdu−dv2=2−3236+2−4236p−12+2−6224p−1+4−6212p−1=576p−444,IRFWG=IRFGM2G=48p−37236p−19,IRAG=∑uvϵEGdu−1/2−dv−1/22=n−2RG=48p−292+3+6p−32+43−56=48p−26+92+43p−106+62+83,IRBG=∑uvϵEGdu1/2−dv1/22=M1DnPn−2RRG=528p−204−2722+483+246p−242+483−126=528p−204−2246+722+483p−246+482+963,IRCG=∑uvϵEGdudvm−2mn=RRGm−2mn=722+483+246p−242+483−12672p−12272p−1248p=46p2+26p+122p2+83p2−42p−83p−36p2+12p−12p6p−1,IRDIFG=∑uvϵEGdudv−dvdu=23−3236+24−4236p−12+26−6224p−1+46−6412p−1=127.9980p−61.9992,IRLG=∑uvϵEGlndu−lndv=ln2−ln336+ln2−ln436p−12+ln2−ln624p−1+ln4−ln612p−1=56.1840p−24.9516,IRLUG=∑uvϵEGdu−dvmindu,dv=2−3236+2−4236p−12+2−6224p−1+4−6412p−190p−48,IRLFG=∑uvϵEGdu−dvdu.dv=2−3636+2−4836p−12+2−61224p−1+4−62412p−1=58.0668p−26.4012,IRLAG=∑uvϵEG2du−dvdu+dv=22−3536+22−4636p−12+22−6824p−1+24−61012p−1=52.7976p−22.3992,IRD1G=∑uvϵEGln1+du−dv=ln1+2−336+ln1+2−436p−12+ln1+2−624p−1+ln1+4−612p−1=131.0064p−42.4776,IRGAG=∑uvϵEGlndu+dv2dudv=ln2+322×336+ln2+422×436p−1+ln2+622×624p−1+ln4+624×612p−1=8.0172p−5.8716.


## 2.2. Irregularity Indices for Metal-Organic NetworkMON2p

In this section, we will compute irregularity indices for the metal-organic networkMON2p. The molecular graph of the metal-organic network MON2p for p=2 is given in Figure 3. It is clear from Figure 3 that there are three types of vertices in the molecular graph of MON2p, i.e., 2, 3, and 4. The cardinality of vertices 2, 3, and 4 is 12p+18, 24p−12, and 12p−6, respectively. The cardinality of the vertex set of MON2p is 48p, i.e., VMON2p=48p. There are five different types of edges present in the molecular graph of MON2p, i.e., 2,3, 2,4, 3,3, 3,4, and 4,4. Their cardinalities are 12p+24, 12p+12, 24p−24, 12p−12, and 12p−12, respectively. The cardinality of the edge set of MON2p is 72p−12, i.e., EMON1p=72p−12.Figure 3
MON2p for p=2.The edge partition of the metal-organic networkMON2p is given in Table 3.Table 3
Partition ofEMON2p.
(du,dv)Frequency(2, 3)12 (p + 2)(2, 4)12 (p + 1)(3, 3)24 (p − 1)(3, 4)12 (p − 1)(4, 4)12 (p − 1)Theorem 2.
EMON2pLet G be the metal-organic network MON2p. The irregularity indices are(1)
VARG=2p2+p−1/4p2(2)
ALG=48p+36(3)
IR1G=624p2+10p−11/p(4)
IR2G=720p−312/48p−72p−12/24p(5)
IRFG=72p+60(6)
IRFWG=72p+60/720p−312(7)
IRAG=32p+16−226+43+32p−86−62+83(8)
IRBG=216p+108−2126+242+243p−486−482+483(9)
IRCG=26p2+46p+43p2+42p2−43p+42p−16p2−8p−1/2p6p−1(10)
IRDIFG=34.9992p+30.9996(11)
IRLG=16.8660p+14.3664(12)
IRLUG=21.9996p+20.0004(13)
IRLFG=16.8468p+14.8188(14)
IRLAG=16.2276p+14.1708(15)
IRD1G=29.8176p+21.5004(16)
IRGAG=1.0740p+1.0716Proof.
Using definitions of irregularity indices given in Table1 and edge partition given in Table 3, we have(3)VARG=∑uεVdu−2mn2=M1Gn−2mn2=456p−13248p−272p−1248p2=2p2+p−14p2,ALG=∑uvϵEGdu−dv=2−312p+24+2−412p+12+3−324p−24+3−412p−12+4−412p−12=48p+36,IR1G=∑uϵVdu3−2mn∑uϵVdu2=FG−2mnM1G=1512p−564−272p−1248p456p−132=624p2+10p−11p,IR2G=∑uvϵEGdudvm−2mn=M2Gm−2mn=720p−31272p−12−272p−1248p=720p−31248p−72p−1224p,IRFG=∑uvϵEGdu−dv2=2−3212p+24+2−4212p+12+3−3224p−24+3−4212p−12+4−4212p−12=72p+60,IRFWG=IRFGM2G=72p+60720p−312,IRAG=∑uvϵEGdu−1/2−dv−1/22=n−2RG=48p−28p−8+32+43+26p+32−43+46=32p+16−226+43+32p−86−62+83,IRBG=∑uvɛEGdu1/2−dv1/22=M1G−2RRG=456p−132−2120p−120+242+243+126p+246−243+242=216p+108−2126+242+243p−486−482+483,IRCG=∑uvϵEGdudvm−2mn=RRGm−2mn=120p−120+242+243+126p+246−243+24272p−12272p−1248p=26p2+46p+43p2+42p2−43p+42p−16p2−8p−12p6p−1,IRDIFG=∑uvϵEGdudv−dvdu=23−3212p+24+24−4212p+12+33−3324p−24+34−4312p−12+44−4412p−12,=34.9992p+30.9996,IRLG=∑uvϵEGlndu−lndv=ln2−ln312p+24+ln2−ln412p+12+ln3−ln324p−24+ln3−ln412p−12+ln4−ln412p−12=16.8660p+14.3664,IRLUG=∑uvϵEGdu−dvmindu,dv=2−3212p+24+2−4212p+24+3−3324p−24+3−4312p−12+4−4412p−12=21.9996p+20.0004,IRLFG=∑uvϵEGdu−dvdu⋅dv=2−3612p+24+2−4812p+12+3−3924p−24+3−41212p−12+4−41612p−12=16.8468p+14.8188,IRLAG=∑uvϵEG2du−dvdu+dv=22−3512p+24+22−4612p+12+23−3624p−24+23−4712p−12+24−4812p−12=16.2276p+14.1708,IRD1G=∑uvϵEGln1+du−dv=ln1+2−312p+24+ln1+2−412p+12+ln1+3−324p−24+ln1+|3−412p−12+ln1+4−412p−12=29.8176p+21.5004,IRGAG=∑uvϵEGlndu+dv2dudv=ln2+322×312p+24+ln2+422×412p+12+ln3+323×324p−24+ln3+423×412p−12+ln4+424×412p−12=1.0740p+1.0716.


## 3. Graphical Comparison

In this section, we give the graphical comparison of results of the metal-organic networksMON1p and MON2p. In Figure 4, the red color is fixed for MON1p and the green color is fixed for MON2p.Figure 4
Graphical comparison of irregularity indices. (a) VAR. (b) AL. (c) IR1. (d) IR2. (e) IRF. (f) IRFW. (g) IRA. (h) IRB. (i) IRC. (j) IRDIF. (k) IRL. (l) IRLU. (m) IRLF. (n) IRLA. (o) IRD1. (p) IRGA.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)(p)


## 4. Conclusion

Topological indices associate a single number with a chemical structure. In quantitative structure activity relationship, knowledge of topological indices plays an important role. In this article, we computed sixteen irregularity indices for metal-organic networksMON1p and MON2p. Most of the calculated topological indices depend on degree-based indices. In the field of chemical graph theory, molecular topology, and mathematical chemistry, a degree-based index also known as a connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. Topological indices are used, for example, in the development of quantitative structure activity relationships (QSARs) in which the biological activity or other properties of molecules are correlated with their chemical structure. Our results are helpful in drug delivery and computer engineering.



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