QSPR Analysis of Certain Graph Theocratical Matrices and Their Corresponding Energy

Vertex Adjacency Matrix: The term vertex adjacency matrix was first introduced in chemical graph theory by Mallion in his interesting paper [20] on graph theocratical aspects of the ring current theory. Below we give the vertex adjacency matrix of the vertex labeled graph G.

et G = (V,E) be a grapLh where V = {1,2,3,··· ,n} the vertex adjacency matrix of a graph with vertex set V = {1,2,3,··· ,n} is the n × n matrix in which a_ij = 1 if and only if there is an adjacency from vertex i to vertex j. Each diagonal entry in the adjacency matrix of a graph is zero. i.e.,

aij={1,if vi is adjacent to vj;0,otherwise.$$\begin{array}{} \displaystyle {a_{{\rm{ij}}}} = \left\{ \begin{array}{*{20}{l}} {1,}&{{\text{if}\;}{v_i}{\;\text{is adjacent to}\;}{v_j};}\\ {0,}&{{\rm{otherwise}}.} \end{array}\right. \end{array}$$

Vertex Zagreb Adjacency Matrix: Motivated by Zagreb matrix [10] we define new adjacency matrix based on the vertex degrees which is as follows:

let G = (V,E) be a graph where V = {1,2,3,··· ,n} then the vertex Zagreb adjacency matrix Z₁(G) of a graph G is defined as follows i.e.,

aij={1,if vi is adjacent to vj;deg(vi)2if vi=vj;0,otherwise.$$\begin{array}{} \displaystyle {a_{ij}} = \left\{ \begin{array}{*{20}{l}} {1,}&{{\text{if}\;}{v_i}{\;\text{is adjacent to}\;}{v_j};}\\ {deg{{\left( {{v_i}} \right)}^2}}&{{\text{if}\;}{v_i} = {v_j};}\\ {0,}&{{\rm{otherwise}}.} \end{array}\right. \end{array}$$

Forgotten Adjacency Matrix: Recently, Gutman and Furtula [7] have studied the forgotten topological index F(G) of a molecular graph G. Based on the definition of adjacency matrix and vertex degrees of a graph G, we define the forgotten adjacency matrix as follows:

let G = (V,E) be a graph where V = {1,2,3,···, n} then the forgotten adjacency matrix F(G) of a graph G is defined as follows i.e.,

aij={1,if vi is adjacent to vj;deg(vi)3if vi=vj;0,otherwise.$$\begin{array}{} \displaystyle {a_{ij}} = \left\{ \begin{array}{*{20}{l}} {1,}&{{\text{if}\;}{v_i}{\;\text{is adjacent to}\;}{v_j};}\\ {deg{{\left( {{v_i}} \right)}^3}}&{{\text{if}\;}{v_i} = {v_j};}\\ {0,}&{{\rm{otherwise}}.} \end{array}\right. \end{array}$$

Harmonic Matrix: Motivated by Harmonic index [6] of a molecular graph G, the harmonic matrix is defined as follows:

let G = (V,E) be a graph where V = {1,2,3,··· ,n} then the harmonic matrix H(G) of a graph G is defined as follows i.e.,

aij={2deg(vi)+deg(vj),if vi is adjacent to vj;0,otherwise.$$\begin{array}{} \displaystyle {a_{ij}} = \left\{ \begin{array}{*{20}{l}} {\frac{2}{{deg({v_i}) + deg({v_j})}},}&{{\text{if}\;}{v_i}{\;\text{is adjacent to}\;}{v_j};}\\ {0,}&{{\rm{otherwise}}{\rm{.}}} \end{array}\right. \end{array}$$

Geometric-Airthmetic Matrix: Again on the same lines of harmonic index, we define GA- matrix [28] based on a GA-index of a molecular graph G as:

let G = (V,E) be a graph where V = {1,2,3,··· ,n} then the harmonic matrix H(G) of a graph G is defined as follows i.e.,

aij={2deg(vi)deg(vj)deg(vi)+deg(vj),if vi is adjacent to vj;0,otherwise.$$\begin{array}{} \displaystyle {a_{ij}} = \left\{ \begin{array}{*{20}{l}} {\frac{{2\sqrt {deg({v_i})deg({v_j})} }}{{deg({v_i}) + deg({v_j})}},}&{{\text{if}\;}{v_i}{\;\text{is adjacent to}\;}{v_j};}\\ {0,}&{{\rm{otherwise}}.} \end{array}\right. \end{array}$$

Degree-Sum Matrix: Ramane et. al [22] have introduced degree-sum matrix associated with a graph and obtained some upper and lower bounds for its eigenvalues.

let G = (V,E) be a graph where V = {1,2,3,··· ,n} then the degree-sum matrix H(G) of a graph G is defined as follows i.e.,

aij={deg(vi)+deg(vj),if vi≠vj.0,otherwise.$$\begin{array}{} \displaystyle {a_{ij}} = \left\{ \begin{array}{*{20}{l}} {deg({v_i}) + deg({v_j}),}&{{\text{if}\;}{v_i} \ne {v_j}.}\\ {0,}&{{\rm{otherwise}}{\rm{.}}} \end{array}\right. \end{array}$$

Laplacian Matrix: In [11] Gutman and Zhou have put forward the Laplacian matrix. The Laplacian matrix sometimes also called a Kirchoff matrix [3] due to its role in matrix tree theorem, Implicitin the electrical network work of Kirchoff in his paper Kirchoff also introduced the concept of the spanning tree.

let G = (V,E) be a graph where V = {1,2,3,··· ,n} then the Laplacian matrix L(G) of a graph G is defined as follows i.e.,

aij={−1,if vi≠vj and vi is adjacent to vj;0if vi≠vj and vi is not adjacent to vj;degvi,vi=vj.$$\begin{array}{} \displaystyle {a_{ij}} = \left\{ \begin{array}{*{20}{l}} { - 1,}&{{\text{if}\;}{v_i} \ne {v_j}{\rm{ and }}{v_i}{\;\text{is adjacent to}\;}{v_j};}\\ 0&{{\text{if}\;}{v_i} \ne {v_j}{\rm{ and }}{v_i}{\rm{ is not adjacent to }}{v_j};}\\ {deg{v_i},}&{{v_i} = {v_j}.} \end{array}\right. \end{array}$$

Sum-Connectivity Matrix: The sum-connectivity matrix denoted by SC_ij was introduced independently by Zhou and Trianjstic [34] it is defined as follows:

let G = (V,E) be a graph where V = {1,2,3,··· ,n} then the harmonic matrix SC_ij(G) of a graph G is defined as follows i.e.,

aij={1deg(vi)+deg(vj),if vi is adjacent to vj;0,otherwise.$$\begin{array}{} \displaystyle {a_{ij}} = \left\{ \begin{array}{*{20}{l}} {\frac{1}{{\sqrt {deg({v_i}) + deg({v_j})} }},}&{{\text{if}\;}{v_i}{\;\text{is adjacent to}\;}{v_j};}\\ {0,}&{{\rm{otherwise}}.} \end{array}\right. \end{array}$$

Vertex Randić Matrix: The vertex-connectivity matrix denoted by Rij introduced by Randic [23]. It can be regarded as edge-weighted matrix of the graph defined as:

let G = (V,E) be a graph where V = {1,2,3,··· ,n} then the harmonic matrix R(G) of a graph G is defined as follows i.e.,

aij={1deg(vi)deg(vj),if vi is adjacent to vj;0,otherwise.$$\begin{array}{} \displaystyle {a_{ij}} = \left\{ \begin{array}{*{20}{l}} {\frac{1}{{\sqrt {deg({v_i})deg({v_j})} }},}&{{\text{if}\;}{v_i}{\;\text{is adjacent to}\;}{v_j};}\\ {0,}&{{\rm{otherwise}}.} \end{array}\right. \end{array}$$

Vertex Adjacency Energy. The characteristic polynomial of A(G) is denoted by

fn(G,a)=det(aI−A(G)).$$\begin{array}{} \displaystyle {f_n}(G,a) = det(aI - A(G)). \end{array}$$

Since A(G) is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order as follows:

a1≥a2≥a3≥⋯,≥an.$$\begin{array}{} \displaystyle {a_1} \ge {a_2} \ge {a_3} \ge \cdots , \ge {a_n}. \end{array}$$

The energy of G is then defined as

E(G)=∑i=1n|ai|.$$\begin{array}{} \displaystyle E(G) = \mathop \sum \limits_n^{i = 1} \left| {{a_i}} \right|. \end{array}$$

Vertex Zagreb Adjacency Energy. The characteristic polynomial of Z₁(G) is denoted by

fn(G,z)=det(zI−Z1(G)).$$\begin{array}{} \displaystyle {f_n}(G,z) = det(zI - {Z_1}(G)). \end{array}$$

Since Z₁(G) is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order as follows:

z1≥z2≥z3≥⋯,≥zn.$$\begin{array}{} \displaystyle {z_1} \ge {z_2} \ge {z_3} \ge \cdots , \ge {z_n}. \end{array}$$

The vertex Zagreb energy of G is then defined as

Z1E(G)=∑i=1n|zi|.$$\begin{array}{} \displaystyle {Z_1}E(G) = \mathop \sum \limits_n^{i = 1} \left| {{z_i}} \right|. \end{array}$$

Forgotten Energy. The characteristic polynomial of F(G) is denoted by

fn(G,f)=det(fI−F(G)).$$\begin{array}{} \displaystyle {f_n}(G,f) = det(fI - F(G)). \end{array}$$

Since F(G) is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order as follows:

f1≥f2≥f3≥⋯,≥fn.$$\begin{array}{} \displaystyle {f_1} \ge {f_2} \ge {f_3} \ge \cdots , \ge {f_n}. \end{array}$$

The forgotten energy of G is then defined as

FE(G)=∑i=1n|fi|.$$\begin{array}{} \displaystyle FE(G) = \mathop \sum \limits_n^{i = 1} \left| {{f_i}} \right|. \end{array}$$

Harmonic Energy. The characteristic polynomial of H(G) is denoted by

fn(G,h)=det(hI−H(G)).$$\begin{array}{} \displaystyle {f_n}(G,h) = det(hI - H(G)). \end{array}$$

Since H(G) is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order as follows:

h1≥h2≥h3≥⋯,≥hn.$$\begin{array}{} \displaystyle {h_1} \ge {h_2} \ge {h_3} \ge \cdots , \ge {h_n}. \end{array}$$

The harmonic energy of G is then defined as

HE(G)=∑i=1n|hi|.$$\begin{array}{} \displaystyle HE(G) = \sum\limits_{i = 1}^n | {h_i}|. \end{array}$$

Geometric-Airthmetic Energy. The characteristic polynomial of GA(G) is denoted by

fn(G,g)=det(gI−GA(G)).$$\begin{array}{} \displaystyle {f_n}(G,g) = det(gI - GA(G)). \end{array}$$

Since GA(G) is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order as follows:

g1≥g2≥g3≥⋯,≥gn.$$\begin{array}{} \displaystyle {g_1} \ge {g_2} \ge {g_3} \ge \cdots , \ge {g_n}. \end{array}$$

The Geometric-Airthmetic energy of G is then defined as

GAE(G)=∑i=1n|gi|.$$\begin{array}{} \displaystyle GAE(G) = \mathop \sum \limits_n^{i = 1} |{g_i}|. \end{array}$$

Degree-Sum Energy. The characteristic polynomial of DS(G) is denoted by

fn(G,d)=det(dI−DS(G)).$$\begin{array}{} \displaystyle {f_n}(G,d) = det(dI - DS(G)). \end{array}$$

Since DS(G) is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order as follows:

d1≥d2≥d3≥⋯,≥dn.$$\begin{array}{} \displaystyle {d_1} \ge {d_2} \ge {d_3} \ge \cdots , \ge {d_n}. \end{array}$$

The Degree-Sum energy of G is then defined as

DSE(G)=∑i=1n|di|.$$\begin{array}{} \displaystyle DSE(G) = \mathop \sum \limits_n^{i = 1} |{d_i}|. \end{array}$$

Laplacian Energy. The characteristic polynomial of L(G) is denoted by

fn(G,λ)=det(λI−L(G)).$$\begin{array}{} \displaystyle {f_n}(G,\lambda ) = det(\lambda I - L(G)). \end{array}$$

Since L(G) is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order as follows:

λ1≥λ2≥λ3≥⋯,≥λn.$$\begin{array}{} \displaystyle {\lambda _1} \ge {\lambda _2} \ge {\lambda _3} \ge \cdots , \ge {\lambda _n}. \end{array}$$

The Laplacian energy of G is then defined as

LE(G)=∑i=1n|λi|.$$\begin{array}{} \displaystyle LE(G) = \mathop \sum \limits_n^{i = 1} |{\lambda _i}|. \end{array}$$

Sum-Connectivity Energy. The characteristic polynomial of SC(G) is denoted by

fn(G,μ)=det(μI−SC(G)).$$\begin{array}{} \displaystyle {f_n}(G,\mu ) = det(\mu I - SC(G)). \end{array}$$

Since SC(G) is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order as follows:

μ1≥μ2≥μ3≥⋯,≥μn.$$\begin{array}{} \displaystyle {\mu _1} \ge {\mu _2} \ge {\mu _3} \ge \cdots , \ge {\mu _n}. \end{array}$$

The Sum-Connectivity energy of G is then defined as

SCE(G)=∑i=1n|μi|.$$\begin{array}{} \displaystyle SCE(G) = \mathop \sum \limits_n^{i = 1} |{\mu _i}|. \end{array}$$

Vertex Randić Energy. The characteristic polynomial of R(G) is denoted by

fn(G,r)=det(rI−R(G)).$$\begin{array}{} \displaystyle {f_n}(G,r) = det(rI - R(G)). \end{array}$$

Since R(G) is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order as follows:

r1≥r2≥r3≥⋯,≥rn.$$\begin{array}{} \displaystyle {r_1} \ge {r_2} \ge {r_3} \ge \cdots , \ge {r_n}. \end{array}$$

The Vertex Randić Energy of G is then defined as

RE(G)=∑i=1n|ri|.$$\begin{array}{} \displaystyle RE(G) = \mathop \sum \limits_n^{i = 1} |{r_i}|. \end{array}$$

Vertex adjacency energyE(G):

bp=−51.397+[E(G)]19.268$$\begin{array}{} \displaystyle bp = - 51.397 + \left[ {E\left( G \right)} \right]19.268 \end{array}$$(2)

mv=75.5727+EG10.1894$$\begin{array}{} \displaystyle mv = 75.5727 + \left[ {E\left( G \right)} \right]10.1894 \end{array}$$(3)

mr=13.1325+[E(G)]3.0530$$\begin{array}{} \displaystyle mr = 13.1325 + \left[ {E\left( G \right)} \right]3.0530 \end{array}$$(4)

hv=10.0231+[E(G)]3.3206$$\begin{array}{} \displaystyle hv = 10.0231 + \left[ {E\left( G \right)} \right]3.3206 \end{array}$$(5)

ct=91.9103+[E(G)]23.1023$$\begin{array}{} \displaystyle ct = 91.9103 + \left[ {E\left( G \right)} \right]23.1023 \end{array}$$(6)

cp=28.9043+[E(G)]0.0858$$\begin{array}{} \displaystyle cp = 28.9043 + \left[ {E\left( G \right)} \right]0.0858 \end{array}$$(7)

st=11.0052+[E(G)]1.1474$$\begin{array}{} \displaystyle st = 11.0052 + \left[ {E\left( G \right)} \right]1.1474 \end{array}$$(8)

mp=−145.3088+[E(G)]4.4911$$\begin{array}{} \displaystyle mp = - 145.3088 + \left[ {E\left( G \right)} \right]4.4911 \end{array}$$(9)

Vertex Zagreb adjacency energyZ₁E(G):

bp=−13.358+[Z1E(G)]4.1092$$\begin{array}{} \displaystyle bp = - 13.358 + \left[ {{Z_1}E\left( G \right)} \right]4.1092 \end{array}$$(10)

mv=94.7149+[Z1E(G)]2.2017$$\begin{array}{} \displaystyle mv = 94.7149 + \left[ {{Z_1}E\left( G \right)} \right]2.2017 \end{array}$$(11)

mr=18.6119+[Z1E(G)]0.6778$$\begin{array}{} \displaystyle mr = 18.6119 + \left[ {{Z_1}E\left( G \right)} \right]0.6778 \end{array}$$(12)

hv=14.0714+[Z1E(G)]0.7822$$\begin{array}{} \displaystyle hv = 14.0714 + \left[ {{Z_1}E\left( G \right)} \right]0.7822 \end{array}$$(13)

ct=211.4016+[Z1E(G)]1.0167$$\begin{array}{} \displaystyle ct = 211.4016 + \left[ {{Z_1}E\left( G \right)} \right]1.0167 \end{array}$$(14)

cp=32.606−[Z1E(G)]0.0988$$\begin{array}{} \displaystyle cp = 32.606 - \left[ {{Z_1}E\left( G \right)} \right]0.0988 \end{array}$$(15)

st=14.4567+[Z1E(G)]0.2108$$\begin{array}{} \displaystyle st = 14.4567 + \left[ {{Z_1}E\left( G \right)} \right]0.2108 \end{array}$$(16)

mp=−139.2218+[Z1E(G)]1.0322$$\begin{array}{} \displaystyle mp = - 139.2218 + \left[ {{Z_1}E\left( G \right)} \right]1.0322 \end{array}$$(17)

Forgotten adjacency energy FE(G):

bp=49.581+[FE(G)]0.829$$\begin{array}{} \displaystyle bp = 49.581 + \left[ {FE\left( G \right)} \right]0.829 \end{array}$$(18)

mv=127.4698+[FE(G)]0.4683$$\begin{array}{} \displaystyle mv = 127.4698 + \left[ {FE\left( G \right)} \right]0.4683 \end{array}$$(19)

mr=29.0759+[FE(G)]0.1390$$\begin{array}{} \displaystyle mr = 29.0759 + \left[ {FE\left( G \right)} \right]0.1390 \end{array}$$(20)

hv=490.0581−[FE(G)]5.8696$$\begin{array}{} \displaystyle hv = 490.0581 - \left[ {FE\left( G \right)} \right]5.8696 \end{array}$$(21)

ct=131.1926+[FE(G)]5.1377$$\begin{array}{} \displaystyle ct = 131.1926 + \left[ {FE\left( G \right)} \right]5.1377 \end{array}$$(22)

cp=31.4563−[FE(G)]0.0249$$\begin{array}{} \displaystyle cp = 31.4563 - \left[ {FE\left( G \right)} \right]0.0249 \end{array}$$(23)

st=18.4178+[FE(G)]0.034156$$\begin{array}{} \displaystyle st = 18.4178 + \left[ {FE\left( G \right)} \right]0.034156 \end{array}$$(24)

mp=−123.9450+[FE(G)]0.2094$$\begin{array}{} \displaystyle mp = - 123.9450 + \left[ {FE\left( G \right)} \right]0.2094 \end{array}$$(25)

harmonic energyHE(G):

bp=−29.71+[HE(G)]32.08$$\begin{array}{} \displaystyle bp = - 29.71 + \left[ {HE\left( G \right)} \right]32.08 \end{array}$$(26)

mv=100.054+[HE(G)]13.9260$$\begin{array}{} \displaystyle mv = 100.054 + \left[ {HE\left( G \right)} \right]13.9260 \end{array}$$(27)

mr=20.4799+[HE(G)]4.2371$$\begin{array}{} \displaystyle mr = 20.4799 + \left[ {HE\left( G \right)} \right]4.2371 \end{array}$$(28)

hv=31.9676+[HE(G)]1.4827$$\begin{array}{} \displaystyle hv = 31.9676 + \left[ {HE\left( G \right)} \right]1.4827 \end{array}$$(29)

ct=118.2041+[HE(G)]38.4066$$\begin{array}{} \displaystyle ct = 118.2041 + \left[ {HE\left( G \right)} \right]38.4066 \end{array}$$(30)

cp=27.5611+[HE(G)]0.4722$$\begin{array}{} \displaystyle cp = 27.5611 + \left[ {HE\left( G \right)} \right]0.4722 \end{array}$$(31)

st=12.1028+[HE(G)]1.9642$$\begin{array}{} \displaystyle st = 12.1028 + \left[ {HE\left( G \right)} \right]1.9642 \end{array}$$(32)

mp=−137.1760+[HE(G)]6.6745$$\begin{array}{} \displaystyle mp = - 137.1760 + \left[ {HE\left( G \right)} \right]6.6745 \end{array}$$(33)

Geometric-Arithmetic energyGAE(G):

bp=−36.99+[GAE(G)]18.984$$\begin{array}{} \displaystyle bp = - 36.99 + \left[ {GAE\left( G \right)} \right]18.984 \end{array}$$(34)

mv=90.04705+[GAE(G)]9.0967$$\begin{array}{} \displaystyle mv = 90.04705 + \left[ {GAE\left( G \right)} \right]9.0967 \end{array}$$(35)

mr=18.0357+[GAE(G)]2.6920$$\begin{array}{} \displaystyle mr = 18.0357 + \left[ {GAE\left( G \right)} \right]2.6920 \end{array}$$(36)

hv=34.6289+[GAE(G)]0.5068$$\begin{array}{} \displaystyle hv = 34.6289 + \left[ {GAE\left( G \right)} \right]0.5068 \end{array}$$(37)

ct=113.4305+[GAE(G)]22.2200$$\begin{array}{} \displaystyle ct = 113.4305 + \left[ {GAE\left( G \right)} \right]22.2200 \end{array}$$(38)

cp=27.3753+[GAE(G)]0.2895$$\begin{array}{} \displaystyle cp = 27.3753 + \left[ {GAE\left( G \right)} \right]0.2895 \end{array}$$(39)

st=12.6092+[GAE(G)]1.0425$$\begin{array}{} \displaystyle st = 12.6092 + \left[ {GAE\left( G \right)} \right]1.0425 \end{array}$$(40)

mp=−133.5245+[GAE(G)]3.28$$\begin{array}{} \displaystyle mp = - 133.5245 + \left[ {GAE\left( G \right)} \right]3.28 \end{array}$$(41)

Degree sum energyDSE(G):

bp=59.931+[DSE(G)]0.897$$\begin{array}{} \displaystyle bp = 59.931 + \left[ {DSE\left( G \right)} \right]0.897 \end{array}$$(42)

mv=138.925+[DSE(G)]0.402892$$\begin{array}{} \displaystyle mv = 138.925 + \left[ {DSE\left( G \right)} \right]0.402892 \end{array}$$(43)

mr=31.1877+[DSE(G)]0.1300$$\begin{array}{} \displaystyle mr = 31.1877 + \left[ {DSE\left( G \right)} \right]0.1300 \end{array}$$(44)

hv=32.5560+[DSE(G)]0.1074$$\begin{array}{} \displaystyle hv = 32.5560 + \left[ {DSE\left( G \right)} \right]0.1074 \end{array}$$(45)

ct=219.9439+[DSE(G)]1.1727$$\begin{array}{} \displaystyle ct = 219.9439 + \left[ {DSE\left( G \right)} \right]1.1727 \end{array}$$(46)

cp=32.1358−[DSE(G)]0.04447$$\begin{array}{} \displaystyle cp = 32.1358 - \left[ {DSE\left( G \right)} \right]0.04447 \end{array}$$(47)

st=18.4122+[DSE(G)]0.04367$$\begin{array}{} \displaystyle st = 18.4122 + \left[ {DSE\left( G \right)} \right]0.04367 \end{array}$$(48)

mp=−122.6285+[DSE(G)]0.2430$$\begin{array}{} \displaystyle mp = - 122.6285 + \left[ {DSE\left( G \right)} \right]0.2430 \end{array}$$(49)

Laplacian energyLE(G):

bp=105.998+[LE(G)]0.795$$\begin{array}{} \displaystyle bp = 105.998 + \left[ {LE\left( G \right)} \right]0.795 \end{array}$$(50)

mv=61.7923+[LE(G)]7.1569$$\begin{array}{} \displaystyle mv = 61.7923 + \left[ {LE\left( G \right)} \right]7.1569 \end{array}$$(51)

mr=9.2778+[LE(G)]2.1462$$\begin{array}{} \displaystyle mr = 9.2778 + \left[ {LE\left( G \right)} \right]2.1462 \end{array}$$(52)

hv=28.1626+[LE(G)]0.7323$$\begin{array}{} \displaystyle hv = 28.1626 + \left[ {LE\left( G \right)} \right]0.7323 \end{array}$$(53)

ct=71.1141+[LE(G)]15.5601$$\begin{array}{} \displaystyle ct = 71.1141 + \left[ {LE\left( G \right)} \right]15.5601 \end{array}$$(54)

cp=31.3140+[LE(G)]0.1221$$\begin{array}{} \displaystyle cp = 31.3140 + \left[ {LE\left( G \right)} \right]0.1221 \end{array}$$(55)

st=10.9055+[LE(G)]0.755115$$\begin{array}{} \displaystyle st = 10.9055 + \left[ {LE\left( G \right)} \right]0.755115 \end{array}$$(56)

mp=−146.7815+[LE(G)]2.8215$$\begin{array}{} \displaystyle mp = - 146.7815 + \left[ {LE\left( G \right)} \right]2.8215 \end{array}$$(57)

Sum connectivity energySCE(G):

bp=−57.214+[SCE(G)]39.634$$\begin{array}{} \displaystyle bp = - 57.214 + \left[ {SCE\left( G \right)} \right]39.634 \end{array}$$(58)

mv=79.5397+[SCE(G)]19.1531$$\begin{array}{} \displaystyle mv = 79.5397 + \left[ {SCE\left( G \right)} \right]19.1531 \end{array}$$(59)

mr=14.5174+[SCE(G)]5.762765$$\begin{array}{} \displaystyle mr = 14.5174 + \left[ {SCE\left( G \right)} \right]5.762765 \end{array}$$(60)

hv=32.3464+[SCE(G)]1.4459$$\begin{array}{} \displaystyle hv = 32.3464 + \left[ {SCE\left( G \right)} \right]1.4459 \end{array}$$(61)

ct=86.5805+[SCE(G)]47.1321$$\begin{array}{} \displaystyle ct = 86.5805 + \left[ {SCE\left( G \right)} \right]47.1321 \end{array}$$(62)

cp=26.5275+[SCE(G)]0.7319$$\begin{array}{} \displaystyle cp = 26.5275 + \left[ {SCE\left( G \right)} \right]0.7319 \end{array}$$(63)

st=0.7399+[SCE(G)]2.3432$$\begin{array}{} \displaystyle st = 0.7399 + \left[ {SCE\left( G \right)} \right]2.3432 \end{array}$$(64)

mp=−144.7607+[SCE(G)]8.7341$$\begin{array}{} \displaystyle \end{array}$$(65)

Vertex Randic energy RE(G):

bp=−33.08+[RE(G)]30.673$$\begin{array}{} \displaystyle bp = - 33.08 + \left[ {RE\left( G \right)} \right]30.673 \end{array}$$(66)

mv=4740.533522−[RE(G)]960.1500$$\begin{array}{} \displaystyle mv = 4740.533522 - \left[ {RE\left( G \right)} \right]960.1500 \end{array}$$(67)

mr=20.0386+[RE(G)]4.0559$$\begin{array}{} \displaystyle mr = 20.0386 + \left[ {RE\left( G \right)} \right]4.0559 \end{array}$$(68)

hv=24.5656+[RE(G)]2.8952$$\begin{array}{} \displaystyle hv = 24.5656 + \left[ {RE\left( G \right)} \right]2.8952 \end{array}$$(69)

ct=116.5371+[RE(G)]36.2135$$\begin{array}{} \displaystyle ct = 116.5371 + \left[ {RE\left( G \right)} \right]36.2135 \end{array}$$(70)

cp=25.3061+[RE(G)]0.9224$$\begin{array}{} \displaystyle cp = 25.3061 + \left[ {RE\left( G \right)} \right]0.9224 \end{array}$$(71)

st=13.07611+[RE(G)]1.6338$$\begin{array}{} \displaystyle st = 13.07611 + \left[ {RE\left( G \right)} \right]1.6338 \end{array}$$(72)

mp=−113.7685+[RE(G)]1.0137$$\begin{array}{} \displaystyle mp = - 113.7685 + \left[ {RE\left( G \right)} \right]1.0137 \end{array}$$(73)