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NANOSYSTEMS: Shashwath S. Shetty, et al. Nanosystems:
PHYSICS, CHEMISTRY, MATHEMATICS Phys. Chem. Math., 2024,15 (3), 315-324.
http://nanojournal.ifmo.ru
Original article DOI 10.17586/2220-8054-2024-15-3-315-324
Energy and spectral radius of Zagreb matrix of graph with applications
Shashwath S. Shetty1", K. Arathi Bhat1b
1Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Karnataka, India [email protected], [email protected] Corresponding author: K. ArathiBhat, [email protected]
Abstract The Z-matrix of a simple graph r is a square symmetric matrix, whose rows and columns correspond to the vertices of the graph and the ijth entry is equal to the sum of the degrees of ith and jth vertex, if the corresponding vertices are adjacent in r, and zero otherwise. The Zagreb eigenvalues of r are the eigenvalues of its Z-matrix and the Zagreb energy of r is the sum of absolute values of its Zagreb eigenvalues. We study the change in Zagreb energy of a graph when the edges of the graph are deleted or rotated. Suppose r is a graph obtained by identifying u e V(r1) and v e V(r2) or adding an edge between u and v, then it is important to study the relation between Zagreb energies of r1, r2 and r. The highlight of the paper is that, the acentric factor of n-alkanes appear to have a strong positive correlation (where the correlation coefficient is 0.9989) with energy of the Z-matrix. Also, the novel correlation of the density and refractive index of n-alkanes with spectral radius of the Z-matrix has been observed.
Keywords spectral radius, energy, Zagreb matrix, acentric factor, density, refractive index Acknowledgements We acknowledge the Institution for overall support. The authors sincerely appreciate the reviewers for all the valuable comments and suggestions, which helped to improve the quality of the manuscript.
For citation Shashwath S. Shetty, K. ArathiBhat Energy and spectral radius of Zagreb matrix of graph with applications. Nanosystems: Phys. Chem. Math., 2024,15 (3), 315-324.
1. Introduction
In mathematical chemistry, graph energies have received considerable attention because of their immense applications [1].Recently, various types of matrices and the corresponding energies of graphs have been studied. In 1978, Gutman defined the energy [2] of a simple graph as the sum of the absolute values of eigenvalues of the adjacency matrix of the corresponding graph.
Let r = (V, E) be an undirected graph on a finite non-empty set of vertices V and the prescribed collection E of an unordered pair of vertices called edges. Two vertices are said to be adjacent, if they share an edge. We use i ~ j to denote that vertices i and j are adjacent. The neighborhood of vertex i in graph r is a collection of all vertices that are adjacent to i in r. The adjacency matrix of a graph r denoted by Ar is n x n matrix whose rows and columns correspond to the vertices of the graphs and the ijth entry is one if vertex i is adjacent to vertex j, and zero otherwise. The energy of a graph is defined as the sum of absolute values of eigenvalues of adjacency matrix of the graph. The singular values of a real matrix A is the square root of the eigenvalues of A1 A, where A1 is the transpose of the matrix A. For a real symmetric positive definite matrix, the singular values of the matrix coincide with the eigenvalues of the matrix. Let ATri is an extended adjacency matrix of the graph r corresponding to the degree based symmetric topological index TI, and A^I [u] be a matrix obtained from A^I by deleting the row and column corresponding to the vertex u, where u is a vertex of the graph r.
The first extended adjacency matrix corresponding to a degree based topological index defined was the Randic matrix [3], and the energy of the corresponding matrix was defined in a similar manner and termed as the Randic energy. The first Zagreb index, M1 (r) of a graph r is defined [4] as the sum of the squares of the degrees over all vertices of the graph. Interesting facts concerned with the first Zagreb index are available in the literature [5,6] and recently it has been considered for graphs with self-loops [7].
The first Zagreb matrix (Z-matrix) [8] of a graph r denoted by Zr is an n x n matrix whose rows and columns correspond to the vertices of the graph and it's ijth entry, zj is given by
«îj
di + dj, if i ~ j ; 0, otherwise,
where dj is the degree of vertex i e V(r). The Zagreb energy of a simple graph r is defined as the sum of the absolute values of the Zagreb matrix eigenvalues of the graph. Let Z1 > Z2 > • • • > Zn be the eigenvalues and Z1 be the Z-spectral radius corresponding to the Z-matrix of the graph r. If x is the principal eigenvector of the Z-matrix then the Rayleigh
© Shashwath S. Shetty, K. Arathi Bhat, 2024
xT 2> x
quotient of Z is a scalar, -and, the supremum value of this quotient over all the vectors x gives one the spectral
xTx
radius of Z, i.e.,
xTZ x Zi = sup T .
x xT x
For all other undefined terminologies of graph theory reader can refer [9].
A few bounds for Z-spectral radius and its variation during deletion and the rotation of an edge is discussed in Section 2. In Section 3, the relation between Zagreb energies of r1; r2 and r is discussed, where r is a graph obtained by identifying u G V(r1) and v G V(r2) or adding an edge between u and v. Chemical applications of Z-spectral radius and Zagreb energy is discussed in the last section.
2. Z-spectral Radius
An edge rotation of e = xy G E(r) around x G V(r), replaces xy by an edge xw where xw G E(r). The variation of Z-spectral radius during the deletion and rotation of an edge is discussed in this section along with few bounds.
Theorem 1. Let r be a connected graph of order n, maximum degree A and Z-spectral radius Z1. If M1 (r) denotes the first Zagreb index of the graph r, then
M1(r)+ £ d2,u
ueV — < Z1 < 2A2,
n
where d2,u is is the sum of degrees of all the vertices which are adjacent to u in R Equality holds in both if and only if V is regular.
Proof. Now,
£ (dU + d2,„) Mi (r)+ £ d2,u
J1Z J ueV (r) ueV (r)
Zi > T1T =-=-,
J1J n n
where d2,u is is the sum of degrees of all the vertices which are adjacent to u in r. Since, the all one vector J is the eigenvector corresponding to the regular graph r, equality holds in the above if and only if r is a regular graph. Let xu is the maximum component of the principal eigenvector x. The eigenvalue equation for the Z-matrix of the simple graph is given by,
Zixu ^ ^ (du + dv )xv •
veN (u)
Therefore,
Zixu = (du + dv )xv < (du + dv )xu
veN(u) veN(u)
< 2A xu
veN(u)
< 2A2 xu.
Equality holds in the above if and only if du = dv = A, i.e., r is regular. □
Theorem 2. [10] Suppose that A = (aij) and B = (bij) are two n x n non-negative symmetric matrices. If A < B, i.e., aij < bij for all i,j, then Zi(A) < Zi(B). Furthermore, if B is irreducible and A = B, then Zi (A) < Zi(B).
Theorem 3. Let r be a forest of order n and size m with n < 2m. Then
2 < Zi(r) < nV2m - n + 1, n
with equality in the lower bound holds if and only if r = ^ K2 and equality in the upper bound holds if and only if r
isomorphic to the star Sn.
Furthermore for n > 2, if r is connected, then
n
6co^-— < Zi(Pn) < Zi(r) n +1
with equality if and only if r = P3
Proof. Since r is a forest, we have di + dj < n for every i, j in V(r). Hence,
Zr < nAr, (1)
where Ar is the adjacency matrix of the graph r. Here, the equality in (1) holds if and only if r is a star graph. Now by using Theorem 2 , Zi < np, where p is the adjacency spectral radius of the graph. But, we have [11] p < %/2m - n + 1 and the equality follows if and only if the graph is a star graph or a complete graph on n vertices.
As n < 2m, we have
with equality if and only if r = ^ K2.
2Ar < Zr
Furthermore, when r is connected and n > 3, we have
3Ar < Zr
with equality if and only if r is a path graph on 3 vertices. We know that,
p(Pn) = 2 cos
n + 1
from which the result follows.
□
Theorem 4. Let r be a connected graph with the Z-spectral radius Z1 and the corresponding principal eigenvector x. If e = uv is not an edge in r, then
Z1(r + e) >Z1(r).
Proof. Let xu denote the uth component of the eigenvector x, corresponding to the spectral radius of r. By using Rayleigh quotient
TZr+ex xTZrx xTC1x
X x x ' ¿j x
Zi(r + e) >--- = —— + _
xTx x-x x-x
where
0n-2xn-2 Ju Jv
Ci = JU 0 d„ + dv +2
JT d„ + dv +2 0
and du and dv are the degrees of the vertices u and v respectively, in the graph r. Ju is the column vector of size (n - 2) corresponding to the vertex u, whose entries are 0 or 1, accordingly the corresponding vertex is adjacent to u or not. Similarly, the column vector Jv corresponds to the vertex v. Therefore
xTC1x = M
xTx xTx
where M = 2(du + dv + 2)xuxv + 2 ^^ xuxw + 2 ^^ xvxy.
w^u y^v
Here M is strictly greater than zero, since x is a positive eigenvector [12].
□
Theorem 5. Let p, q, r be the vertices of a connected graph r such that rp is an edge in r, where as rq is not an edge in r. Then
Zi(r - rp + rq) > Zi(r), if N > 0
and the inequality is strict if N > 0,
where N = dr (xq — xp)xr + (dqxq + xq — dpxp)xr + ^ xqXy — ^ xpxw. Proof. Let x be the eigenvector corresponding to Z1 (r). Then, we have
Zi (r — rp + rq) >
CTJ7 (r-rp+rq), x-x
TZrx xTC2x
x-x
+
where
2=
Jn—3xn-
>1xn-3
_ J
J p
xTx
Jq
—J- —(dr + dp)
'p q
(dr + dp) dr + dq + 1 0 0
JT
dr + dq + 1
0
0
where dp, dq and dr are the degrees of the vertices p, q and r respectively, in the graph r. Jp is the column vector of size (n - 3) corresponding to the vertex p, whose entries are 0 or 1, accordingly the corresponding vertex is adjacent to p or not. Similarly, the column vector Jq corresponds to the vertex q. Hence
xTC2x = 2N
xTx
xTx
where 2N = 2(dr + dq + 1)xrxq — 2(dr + dp)xrxp + 2 ^ xqxy — 2 ^ xpxw.
q y / ^p^w■
y~q,y = r w~p,w = r
□
n
0n-3x1
0
3. Zagreb energy
The coalescence of two graphs r and r2 denoted by r o r2 is obtained by identifying two arbitrary vertices, u e V(ri) and v e V(r2). The change in adjacency energy during the coalescence of two graphs was studied in [13]. Similar type result is discussed below for the Zagreb energy.
Theorem 6. Let r1 o r2 be coalescence of two graphs r1 and r2 obtained by identifying the vertices u e V (r1) and v e V(r2). Then
ZE(ri or2) < ZE(ri) + ZE(r2) + + 2dv\fd~u
with equality if and only if u is an isolated vertex in r1 and/or v is an isolated vertex in r2.
Proof. It is direct that
Z
rior2 _
Zri [u] [xi + x2]T 0
[xi + X2] 0
[yi + y2]
0
[yi + y2]T
Zr2 [v]
where x1 is the deleted column of Zri corresponding to the vertex u and x2 is the column vector obtained by replacing all the non-zero entries of x1 by dv. Similarly, y2 is the deleted column of Zr2 corresponding to the vertex v and y1 is the column vector obtained by replacing all the non-zero entries of y2 by du. Now,
Zrior2 = A1 + A2,
where
Ai
A
Zri [u] [xi + X2] [xi + X2]T 0
00
and
0 0
0
2 = u u [yi + y2]7 0 [yi + y2] Zr2 [v]
For a symmetric matrix singular values are nothing but the absolute values of the eigenvalues and hence by using the Ky Fan's inequality [13] for singular values of the matrix, we have
E (Zrior2 ) < E (Ai)+ E (A2) (2)
and the equality holds if and only if there exists an orthogonal matrix Q such that both QA1 and QA2 are positive semi-definite. Since Q is orthogonal,
Q11 Q21 Q31 Q11 Q12 Q13
QTQ = Q12 Q22 Q32 Q21 Q22 Q23 = I,
Q13 Q23 Q33 Q31 Q32 Q33
where the blocks Qij's are of appropriate size. Therefore
Q11Q11 + Q2iQ2i + Q5i Qai
Q13Q13 + Q2aQ23 + QJa Qaa
QiiQia + Q2iQ2a + QJiQaa
I, I,
0.
Consider,
QAi
Q11 Zri [u] + Q 12 [xi + x2]T Q21Zri [u] + Q22 [xi + x2]T Q31Zri [u] + Q32[xi + x2]T
Q11 [xi + x2] Q21 [xi + x2] Q31[xi + x2]
which implies Q31 [xi + x2] = 0. Also,
QA2 =
0 Q13[yi + y2] Q12[yi + y2]T + Q13Zr2 [v] 0 Q23 [yi + y2] Q22[yi + y2]T + Q23Zr2 [v] 0 Q33[yi + y2] Q32[yi + y2]T + Q33Zr2 [v]
which implies Qi3[yi + y2] = 0. Hence,
[Qii[xi + X2 ]]TQi3[yi + y2] + [Q2i[xi + x2 ]]TQ23[yi + y2] + [Q3l[xi + X2]]TQ33[yi + y2] = 0
implies
[Q2i[xi + X2]]TQ23[yi + y2] =0, and here both Q2i [xi + x2] or Q23[yi + y2] are scalars.
Now if Q2i [xi + x2] = 0, and since QAi is diagonally dominant we arrive at [xi + x2] = 0, which implies that u is an isolated vertex in ri. Similarly if Q23 [yi + y2] = 0, we arrive at [yi + y2] =0, which implies that v is an isolated vertex in r2. Let
Ai = A3 + A4 and A2 = A5 + Ae
where
zri [u] xi 0 0 x2 0
A3 = xT 0 0 , A4 = xT 0 0
0 0 0 0 0 0
0 0 0 0 0 0
A5 = 0 0 yT and Ae = 0 0 yT
0 y2 Zr2 [v] 0 yi 0
Again by using the same inequality,
E(Ai) < E(A3) + E(A4)
= ZE (ri) + 2^X2 = ZE (ri) +
and the equality holds if and only if either u is an isolated vertex in r1 or v is an isolated vertex in r2. Similarly,
E(A2) <E(As) + E(Ae)
= ze (r2) + = ze (r2) +
with equality condition as stated above. □
The change in Zagreb energy by adding an edge between the two vertices of two different graphs is given below.
Theorem 7. Let r be a graph obtained by adding a bridge e = uv between the vertices u e V (r1) and v e V (r2) and let d'u and d'v be the degrees of the vertices u and v in the original graphs r1 and r2 respectively. Then
ZE (r) < ZE (ri) + ZE (r2) + 2^v/dU + VdV + d' + d'v + 2)
and the equality holds if and only if u and v are isolated vertices in r.
Proof. The Z-matrix corresponding to the above mentioned graph r is,
Zri [u] [xi + Ju] 0 0
[xi + Ju]T 0 d'u + d' +2 0
0 d' + dV + 2 0 [yi + Jv ]T
0 0 [yi + Jv ] Zr2 [v]
Here xi (and yi) represents the column vector of Zri (and Zr2) corresponding to the vertex u (and v) in the respective graph. The column vector Ju is obtained by replacing all the non-zero entries of xi by 1 and, the column vector Jv is obtained by replacing all the non-zero entries of yi by 1.
Z1 =
Z1
Bi + B2
where
Si
zri [u] xi 0 0
xT 0 0 0
0 0 0 yT
0 0 yi Z r2
and
B-
2 —
0 JM 0 0
JU 0 dU + dV + 2 0
0 dU + dV +2 0 JT
0 0 JV 0
Hence by using the singular values inequality,
Now, it is easy to observe that
Let
where
ZE(r) <E(Bi)+ E(B2). E (Bi) — ZE (ri)+ ZE (r2).
B2 — B3 + B4,
(3)
0 JU 0 0
JU 0 0 0
0 0 0 JT
0 0 JV 0
and
B.
4—
0 d„ + d; +2 0 0
Therefore by applying the same inequality again, we get
E(B2) < E(B3) + E(B4) = ^v/dU +
0 0
0
dU + dV +2 0 0
+ 2(dU + dV +2).
(4)
Equality holds in the inequality (4), if and only if there exists an orthogonal matrix Q, such that both QB3 and QB4 are positive semi-definite. If we assume the existence of such an orthogonal matrix Q, we have
QTQ
11 Q21 Q31 Q41
12 Q22 Q32 Q42
13 Q23 Q33 Q43
14 Q24 Q34 Q44
Q11 Q21 Q31
Ql2 Q22
Q32
Q13 Q23
Q33
Q14 Q24
Q34
Q41 Q42 Q43 Q44
where Qij, 1 < i, j < 4, are assumed to have the appropriate size. Hence,
Q12Q12 + Q22Q22 + Q32Q32 + Q42Q42 = Ij
Q13Q13 + Q33Q23 + Q33Q33 + Q23Q43 = I,
QB4
0 Q13K + dV + 2) Q12(dU + dV +2) 0
0 Q23K + dV + 2) Q22(dU + dV +2) 0
0 Q33K + dV + 2) Q32(dU + dV +2) 0
0 Q43(dU + dV + 2) Q42(dU + dV +2) 0
v
Since QB4 is diagonally dominant, we have Q13 So Equation (5) reduces to
Q12 = Q43 = Q42 = 0, as d« + dV +2 can not be equal to zero.
QLQ22 + QLQ32 = I,
Q23Q23 + Q33Q33 = I-
(6)
Also,
QB3
Q12JU Ql1 J-U Q14Jv Q13JU
Q22JU Q2I J-U Q24Jv Q23JU
Q32JU Q31JM Q34Jv Q33JU
Q42JU Q4I J-U Q44Jv Q43JU
0 Q11JM Q14Jv 0
Q22JU Q2I J-U Q24Jv Q23JU
Q32JU Q31JM Q34Jv Q33JU
0 Q41JM Q44Jv 0
Again, Q11J dominant.
Now, from Equation (6),
Q14 Jv = Q22 J« = Q32 J« = Q41 J« = Q44JV = Q33 J« = Q23JU = 0, as QB3 is also diagonally
Ju
J«Q22Q22 + J«Q32Q32 = [Q22J«]TQ22 + [Q32J«]TQ32 = 0, JV — Jv Q23Q23 + Jv Q33Q33
= [Q23JT]TQ23 + [Q33JT]TQ33 = 0, which implies both u and v are isolated vertices in r and ri respectively.
Now, it is direct that equality in (3) holds only if the equality in (4) holds and it is direct that, the equality in (3) holds when u and v are isolated vertices in r1 and r2 respectively. Hence the result follows. □
The energy of a graph and a subgraph obtained by deleting an edge is discussed below.
Theorem 8. Let r be a graph and e = uv be an edge in r and let d« and dv be the degrees of the vertices u and v in the resultant graph r — e. Then
ZE(r — e) >ZE(r) — 2(v/d« + v^ + d« + dv +2) uv is an isolated edge in r.
with equality if and only if e Proof. Consider
Z1 =
Z r[u,v]
[xi + X2 ]T
[yi + y2]T
[xi + X2 ] 0
d„ + dv + 2
[yi + y2 ]
d„ + dv + 2 0
where x1 and y1 represents the column vector of Zr corresponding to the vertex u and v. The column vector x2 is obtained by replacing all the non-zero entries of x1 by —1 and, the column vector y2 is obtained by replacing all the non-zero entries of y1 by —1.
The proof follows similarly. □
4. Applications
Pi-electron energy is an important concept in chemistry and material science. It is crucial in describing electron interactions in the pi-orbitals of adjacent atoms in conjugated systems such as double bonds and aromatic rings. These pi-electrons' energy levels have an impact on the stability, reactivity, and electronic properties of organic compounds and conjugated materials. By using the Huckel molecular orbital (HMO) theory one can get many important properties of the conjugated molecules using the total pi-electron energy.
For vinyl compounds specifically, which are organic compounds containing the vinyl functional group (R-CH = CH2), the Total pi-electron energy would quantify the energy associated with the pi-electrons within the double bond. The Zagreb indices would describe the molecular size and branching patterns of the compound's structure. The pi-electron energy of the polyenes and vinyl compounds [14] are compared with the Zagreb energy and also the scatter plot
is shown in Fig. 1, which shows that the Zagreb energy and pi-electron energy are highly correlated with the correlation coefficient of 0.9971.
Fig. 1. Scatter plot of total pi-electron energy v/s Z-energy of polyenes and vinyl compounds.
The acentric factor is directly related to the critical properties of a substance, such as its critical temperature and critical pressure. R. Zheng et al. [15], have compared the experimental results of entropy and acentric factor of octane isomers (Fig. 2) with the spectral radius of Arithmetic-Geometric (AG), Atom Bond Connectivity (ABC) and Sombor (S) matrix of the molecular graphs of the same. For octane isomers, they found that correlation coefficient of the spectral radius of AG, ABC and S with entropy are -0.917, -0.906 and -0.912, respectively, and with acentric factor it is found to be -0.947 - 0.930 and -0.962, respectively.
The acentric factor depends on molecular shape and size, which are influenced by factors such as molecular weight,
branching, and symmetry. Alkanes with higher molecular weight and more branching tend to have higher acentric factors
because they deviate more from spherical shape and have greater molecular interactions. Entropy is a measure of the
number of microscopic configurations or arrangements that are consistent with the macroscopic state of a system. In
general, increasing molecular branching tends to increase entropy. This is because branching increases the number of
possible arrangements of the molecules, leading to a greater number of microstates accessible to the system. Contrary
to this, the spectral radius of graphs (in general) decreases if they deviate more from spherical shape. That is, star graph
__n
has the maximum spectral radius (Vn - 1) and path graph has the minimum (2 cos-). The correlation study of
n +1
thermodynamic properties of aromatic compounds with the various topological indices obtained from the eigenvalues of the adjacency matrix has been carried out in [16]. The present study shows that, Z-spectral radius and the acentric factor (entropy) of the octane isomers are negatively correlated with the correlation coefficient of -0.974 (-0.9168). The experimental values of the entropy and acentric factor of octane isomers were taken from [15].
Fig. 2. Octane isomers
Since these factors are well correlated, we can predict the acentric factor and entropy of the octane isomers. Now, from Figs. 6 and 7 we have
acenfac = -0.02172 x Zi + 0.5597, entropy = -2.606 x Zi + 132.3. The experimental values of density (in g/cm3), refractive index and acentric factors of the n-alkanes from C2 to C30 are taken from [17] and from [18-20] and also the Z-spectral radius and Z-energy of the molecular graph of the same has
Fig. 3. Linear fit for the scatter plot of acenfac v/s Z-spectral radius.
Fig. 4. Linear fit for the scatter plot of entropy v/s Z-spectral radius.
Fig. 5. Scatter plot of acentric factor against Zagreb energy.
Fig. 6. Scatter plot of density (in gcm-3) against Zagreb spectral radius.
Fig. 7. Scatter plot of refractive index against Zagreb spectral radius.
been computed. Previously, it is observed that the acentric factors of octane isomers are negatively correlated to the Z-spectral radius. Now, a very high positive correlation between the Z-energy and the acentric factor of n-alkanes has been observed, where the correlation coefficient is found to be 0.9989. Even though the branching is similar, the number of carbon atoms are different and more the number of carbon atoms higher the Zagreb energy (of its graphical representation) and also the acentric factor. This increase in acentric factor is because, longer the carbon chain in n-alkanes (i.e., as n increases), more will be the deviation from the spherical shape.
When a molecule is highly branched, it tends to have a lower density compared to a molecule with a linear or less-branched structure. This lower density can lead to a decrease in the refractive index of the material. Hence as expected, it is found that, density and refractive index of the n-alkanes are positively correlated to Z-spectral radius, with respective correlation coefficient of 0.9821 and 0.9693. Figs.5,6 and 7 show that one can predict the acentric factor (and hence the critical properties), density and refractive index of n-alkanes, just by computing the Z-spectral radius and Z-energy.
Conclusion
in this paper, we found some bounds for the spectral radius of the Zagreb matrix of graphs and also we study how it changes during the deletion and rotation of an edge. Also, the change in Zagreb energy of a graph, which is obtained by deleting an edge or identifying the end vertices of an edge, are studied. Mainly, the excellent correlation between the acentric factor (density, refractive index) of n-alkanes and the Zagreb energy (and spectral radius) of the molecular graph of the same has been observed.
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Submitted 23 January 2024; revised 30 April 2024; accepted 11 May 2024
Information about the authors:
Shashwath S. Shetty - Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Karnataka, 576104, India; ORCID 0009-0004-0324-1835; [email protected]
K. Arathi Bhat - Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Karnataka, 576104, India; ORCID 0000-0002-1526-5760; [email protected]
Conflict of interest: the authors declare no conflict of interest.
