EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 2, 2023, 1094-1109
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Semi-Total Point Graph of Neighbourhood Edge
Corona Graph OF G and H

Ika Hesti Agustin3,4, A. S. Maragadam2, Dafik1,4,∗, V. Lokesha2, M. Manjunath2

1 Department of Mathematics Education, University of Jember, Indonesia
2 Department of Mathematics, V. S. K. University, Vinayaka Nagara, Ballari, India
3 Department of Mathematics, University of Jember, Indonesia
4 PUI-PT Combinatorics and Graph, CGANT, University of Jember, Indonesia

Abstract. A topological index is a function having a set of graphs as its domain and a set of
real numbers as its range. Here we concentrated on topological indices involving the number of
vertices, the number of edges and the maximum and minimum vertex degree. The aim of this paper
is to compute the lower and upper bounds of the second Zagreb index, third Zagreb index, Hyper
Zagreb index, Harmonic index, Redefined first Zagreb index, First reformulated Zagreb index,
Forgotten topological index, square F -index, Sum-connectivity index, Randic index, Reciprocal
Randic index, Gourava index, Sombar index, Nirmala index, Geometric-Arithmetic index and
lower bonds of Atom bond connectivity index, Redefined second Zagreb.

2020 Mathematics Subject Classifications: AMS 05C05, 05C90, 05C12.

Key Words and Phrases: Semi-Total point Graph, Corona Product of Graphs, Neighborhood
Edge Corona Graph, Lower and Upper Bounds of Topological indices.

1. Introduction

A graph invariant that correlates the physico-chemical properties of a molecular graph
with a number is called a topological index. The first topological index was introduced by
Wiener, a chemist, in 1947 to calculate the boiling points of paraffins [22]. Applications
of molecular structure descriptors are a standard procedure in the study of structure -
property relations nowadays, especially in the field of QSPR/QSAR study [13],[27],[24],
and [16]. During the last century, theoretical chemists started working on the use of
topological indices to obtain information of various properties of organic substances which
depend upon their molecular structure. For this purpose, numerous topological indices
were found and studied in the chemical literature. Throughout the paper, we only consider

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4513

Email addresses: maragadamvijay@gmail.com (A. S. Maragadam), d.dafik@unej.ac.id (Dafik),
v.lokesha@gmail.com (V. Lokesha), manju347@gmail.com. ( M. Manjunath)

https://www.ejpam.com 1094 © 2023 EJPAM All rights reserved.



Dafik et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1094-1109 1095

simple graphs without isolated vertex. The study on topological indices and its properties
are growing vastly, thus many results can be found, for instance in [10],[21],[26], and [20].

Now we recall some well known topological indices. For a graph G, Gutman et al.
[12] defined the second Zagreb index as M2(G) =

∑
uv∈E(G)[du · dv]. Fath-Taber et al. [9]

proposed the third Zagreb index and defined it as ZG3(G) =
∑

uv∈E(G) |du − dv|. The

Hyper Zagreb index is defined in [23] as M [G] =
∑

uv∈E[G][du+dv]
2. The Harmonic index

is defined in [14] as H[G] =
∑

uv∈E[G]
2

du+dv
. The Redefined first Zagreb index is defined

in [4] as ReZG1[G] =
∑

uv∈E[G]

[
du+dv
du·dv

]
. The First Reformulated Zagreb index is defined

in [19] as EM1[G] =
∑

uv∈E[G][du + dv − 2]2

Furthermore, for a graph G, Furtula et al., [8] proposed the definition of the Forgotten
topological index as F (G) =

∑
uv∈E(G)[d

2
u + d2v]. The extension works which have to

be done on this topological indices are recommended in [17][28].The square F -index of a
graph G is defined in [30] as QF [G] =

∑
uv∈E[G][d

2
u − d2v]

2. The Sum-connectivity index is
defined in [2] as

SC[G] =
∑

uv∈E[G]

2√
du + dv

The Randic index is defined in [18] as

R[G] =
∑

uv∈E[G]

1√
dudv

Moreover, for a graph G, the Reciprocal Randic index is defined in [3] as RR[G] =∑
uv∈E[G]

√
du.dv. Gourava index of graph G is defined in [29] as GO(G) =

∑
uv∈E(G)[du+

dv + dudv]. The Atom bond connectivity index is defined in [7] as

ABC[G] =
∑

uv∈E[G]

√
du + dv − 2

du.dv

The Redefined second Zagreb index is defined in [5] as

ReZG2[G] =
∑

uv∈E[G]

du · dv
du + dv

The Geometric-Arithmetic index is defined in [6] as

GA[G] =
∑

uv∈E[G]

2
√
du.dv

du + dv

The Sombor index is defined in [11] as SO[G] =
∑

uv∈E[G]

√
d2u + d2v. Finally, the Nirmala

index is defined in [31] as N [G] =
∑

uv∈E[G]

√
du + dv.

In the following, we will recall the two definitions which are important in this paper.



Dafik et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1094-1109 1096

Definition 1. [15] The semi-total point graph of neighbourhood edge corona graph of G
and H is a connected graph, denoted by G⊖nR H = ψ.

Definition 2. [25] By G⊖nR H = ψ, we mean a graph obtained from one copy of R(G)
and m1 copies of H and joining every vertex of ith copy of H to the vertices which are
incident to the edge ei ∈ E(G), [1 ≤ i ≤ m1] .

Throughout the paper, we utilize the finite simple connected graphs. Let G and H
be graphs with vertex sets V (G), V (H) and edge sets E(G), E(H), respectively. The
degree of vertex v is the number of vertices adjacent to v. Let {V (G) ∩ V (H) = ∅/g ∈
V (G), h ∈ V (H)}. The number of vertices and number of edges in the graphs G and H are
represented by v1 , v2 and e1 , e2, respectively. By this definition, we have ∆G ≥ degG(g),
and δG ≤ degG(g) The bounds for different topological indices are obtained by many
researchers for graphs.

Now we will define a new class of graph operator , namely semi-total point graph of
neighbourhood edge corona graph of G and H as (ψ graph), see [1].

Definition 3. By G⊖nRH = ψ, we mean a graph obtained from one copy of graph G and
e1 copies of H and joining a vertex of V (G), that is, on the ith vertex in G is adjacent to
every vertex of ith copy of H.



Dafik et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1094-1109 1097

Table 1. Edge partition of ψ graph
Edge dG(2 + v2), dG(2 + v2) (2, dG(2 + v2) (dH + 2, dH + 2) (dH + 2, dG(2 + v2)

frequency e1 2e1 e1e2 2v2e1

From now on, we will ready to describe our new results related to bounds for defined
class of graphs using recalled topological indices.

2. Bounds on various topological indices of ψ graph

In this section, we formulate the bounds on the M2, ZG3, HM , H, ReZG1, EM1, F ,
QF , SC, R, RR, GO, ABC, ReZG2, GA, SO and N indices of ψ graph.

Theorem 1. Let G and H be two simple connected graphs, then

ZG3[ψ] ≤ 2e1|2− 2∆G −∆Gv2|+ 2v2e1|∆G + 2− 2∆G − v2∆G|

and
ZG3[ψ] ≥ 2e1|2− 2δG − δGv2|+ 2v2e1|δG + 2− 2δG − v2δG|.

Proof. Using Table 1 and the definition of third Zagreb index, we have the following

ZG3(ψ) =
∑

uv∈E(G)

|du − dv|

= e1|dG(2 + v2)− dG(2 + v2)|+ 2e1|2− dG(2 + v2)|+ e1e2|(dH + 2)− (dH + 2)|
+ 2v2e1|(dH + 2)− dG(2 + v2)|

= e1|0|+ 2e1|2− 2dG + dGv2|+ e1e2|0|+ 2v2e1|dH + 2− 2dG + dGv2|
= 2e1|2− 2dG − dGv2|+ 2v2e1|dG + 2− 2dG − v2dG|

Z3[ψ] ≤ 2e1|2− 2∆G −∆Gv2|+ 2v2e1|∆G + 2− 2∆G − v2∆G|.

Similarly, we have Z3[ψ] ≥ 2e1|2− 2δG − δGv2|+ 2v2e1|δG + 2− 2δG − v2δG|.

Theorem 2. Let G and H be two simple connected graphs. We have

F [ψ] ≤ e1|2∆2
G(2+v2)

2|+2e1|4+∆2
G(2+v2)

2|+e1e2|2(∆H+2)2|+2v2e1|(∆H+2)2+∆2
G(2+v2)

2

and

F [ψ] ≥ e1|2δ2G(2+v2)2|+2e1|4+δ2G(2+v2)2|+e1e2|2(δH+2)2|+2v2e1|(δH+2)2+δ2G(2+v2)
2|.

Proof. Using Table 1 and the definition of forgotten topological index, we have

F (ψ) =
∑

uv∈E(G)

|d2u + d2v|

= e1|d2G(2 + v2)
2 + d2G(2 + v2)

2|+ 2e1|22 + d2G(2 + v2)
2|+ e1e2|(dH + 2)2 + (dH + 2)2|



Dafik et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1094-1109 1098

+ 2v2e1|(dH + 2)2 + d2G(2 + v2)
2|

= e1|2d2G(2 + v2)
2|+ 2e1|4 + d2G(2 + v2)

2|+ e1e2|2(dH + 2)2|
+ 2v2e1|(dH + 2)2 + d2G(2 + v2)

2|
F (ψ) ≤ e1|2∆2

G(2 + v2)
2|+ 2e1|4 + ∆2

G(2 + v2)
2|+ e1e2|2(∆H + 2)2|

+ 2v2e1|(∆H + 2)2 +∆2
G(2 + v2)

2|

Similarly,

F [ψ] ≥ e1|2δ2G(2+v2)2|+2e1|4+δ2G(2+v2)2|+e1e2|2(δH+2)2|+2v2e1|(δH+2)2+δ2G(2+v2)
2|.

Theorem 3. Let G and H be two simple connected graphs, then

GO[ψ] ≤ e1[4∆G + 2v2∆G +∆2
G(2 + v2)

2] + 2e1[6∆G + 3v2∆G + 2] + e1e2[2∆H + 4

+ (∆H + 2)2] + 2v2e1[2∆G +∆H +∆Gv2 + 2 +∆G(∆H + 2)(2∆G + v2)]

and

GO[ψ] ≥ e1[4δG + 2v2δG + δ2G(2 + v2)
2] + 2e1[6δG + 3v2δG + 2] + e1e2[2δH + 4

+ (δH + 2)2] + 2v2e1[2δG + δH + δGv2 + 2 + δG(δH + 2)(2δG + v2)].

Proof. Using Table 1 and definition of Gourava index, we have

GO(ψ) =
∑

uv∈E(G)

[du + dv + dudv]

= e1[dG(2 + v2) + dG(2 + v2) + d2G(2 + v2)
2] + 2e1[2 + dG(2 + v2) + 2dG(2 + v2)]

+ e1e2[(dH + 2) + (dH + 2) + (dH + 2)2] + 2v2e1[(dH + 2) + dG(2 + v2)

+ dG(dH + 2)(2dG + v2)]

GO(ψ) ≤ e1[4∆G + 2v2∆G +∆2
G(2 + v2)

2] + 2e1[6∆G + 3v2∆G + 2] + e1e2[2∆H + 4

+ (∆H + 2)2] + 2v2e1[2∆G +∆H +∆Gv2 + 2 +∆G(∆H + 2)(2∆G + v2)].

Similarly,

GO[ψ] ≥ e1[4δG + 2v2δG + δ2G(2 + v2)
2] + 2e1[6δG + 3v2δG + 2] + e1e2[2δH + 4 + (δH + 2)2]

+ 2v2e1[2δG + δH + δGv2 + 2 + δG(δH + 2)(2δG + v2)].

Theorem 4. Let G and H be two simple connected graphs, then

M2[ψ] ≤ e1[∆
2
G(2+ v2)

2]+ 4e1[∆G(2+ v2)]+ e1e2[(∆H +2)2]+ 2v2e1[(∆H +2)∆G(2+ v2)]

and

M2[ψ] ≥ e1[δ
2
G(2 + v2)

2] + 4e1[δG(2 + v2)] + e1e2[(δH + 2)2] + 2v2e1[(δH + 2)δG(2 + v2)].



Dafik et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1094-1109 1099

Proof. Using table 1 and definition of second zagreb index, we have

M2(ψ) =
∑

uv∈E(G)

[dudv]

= e1[(dG(2 + v2))(dG(2 + v2))] + 2e1[2dG(2 + v2)] + e1e2[(dH + 2)(dH + 2)]

+ 2v2e1[(dH + 2)dG(2 + v2)]

= e1[d
2
G(2 + v2)

2] + 4e1[dG(2 + v2)] + e1e2[(dH + 2)2] + 2v2e1[(dH + 2)dG(2 + v2)]

M2[ψ] ≤ e1[∆
2
G(2 + v2)

2] + 4e1[∆G(2 + v2)] + e1e2[(∆H + 2)2] + 2v2e1[(∆H + 2)∆G(2 + v2)].

similarly, M2[ψ] ≥ e1[δ
2
G(2+v2)

2]+4e1[δG(2+v2)]+e1e2[(δH +2)2]+2v2e1[(δH +2)δG(2+
v2)].

Theorem 5. Let G and H be two simple connected graphs, then

QF (ψ) ≤ 2e1[4−∆2
G(2 + v2)

2] + 2v2e1[(∆H + 2)2 −∆2
G(2 + v2)

2]

and
QF (ψ) ≥ 2e1[4− δ2G(2 + v2)

2] + 2v2e1[(δH + 2)2 − δ2G(2 + v2)
2].

Proof. Using Table 1 and definition of square F -index, we have

QF [ψ] =
∑

uv∈E[G]

[d2u − d2v]
2

= e1[d
2
G(2 + v2)

2 − d2G(2 + v2)
2] + 2e1[2

2 − d2G(2 + v2)
2] + e1e2[(dH + 2)2 − (dH + 2)2]

+ 2v2e1[(dH + 2)2 − d2G(2 + v2)
2]

= 2e1[4− d2G(2 + v2)
2] + 2v2e1[(dH + 2)2 − d2G(2 + v2)

2]

QF [ψ] ≤ 2e1[4−∆2
G(2 + v2)

2] + 2v2e1[(∆H + 2)2 −∆2
G(2 + v2)

2].

Similarly, QF (ψ) ≥ 2e1[4− δ2G(2 + v2)
2] + 2v2e1[(δH + 2)2 − δ2G(2 + v2)

2].

Theorem 6. Let G and H be two simple connected graphs, then

EM1(ψ) ≤ e1[2∆G(2+v2)−2]2+2e1[∆G(2+v2)]
2+e1e2[2∆H+2]2+2v2e1[∆H+∆G(2+v2)]

2

and

EM1(ψ) ≥ e1[2δG(2+v2)−2]2+2e1[δG(2+v2)]
2+e1e2[2δH +2]2+2v2e1[δH +δG(2+v2)]

2

Proof. Using Table 1 and the definition of first reformulated zagreb index, we have

EM1[ψ] =
∑

uv∈E[G]

[du + dv − 2]2

= e1[dG(2 + v2) + dG(2 + v2)− 2]2 + 2e1[2 + dG(2 + v2)− 2]2



Dafik et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1094-1109 1100

+ e1e2[dH + 2 + dH + 2− 2]2 + 2v2e1[dH + 2 + dG(2 + v2)− 2]2

+ 2v2e1[(dH + 2)2 − d2G(2 + v2)
2]

EM1(ψ) ≤ e1[2∆G(2 + v2)− 2]2 + 2e1[∆G(2 + v2)]
2 + e1e2[2∆H + 2]2

+ 2v2e1[∆H +∆G(2 + v2)]
2

similarly,

EM1(ψ) ≥ e1[2δG(2 + v2)− 2]2 + 2e1[δG(2 + v2)]
2 + e1e2[2δH + 2]2 + 2v2e1[δH

+ δG(2 + v2)]
2.

Theorem 7. Let G and H be two simple connected graphs, then

HM [ψ] ≤ e1[2∆G(2 + v2)]
2 + 2e1[2 + ∆G(2 + v2)]

2 + e1e2[(∆H + 2)2]

+ 2v2e1[∆H + 2 +∆G(2 + v2)]
2

and

HM [ψ] ≥ e1[2δG(2 + v2)]
2 + 2e1[2 + δG(2 + v2)]

2 + e1e2[(δH + 2)2]

+ 2v2e1[δH + 2 + δG(2 + v2)]
2.

Proof. Using Table 1 and definition of Hyper zagreb index, we have

HM(ψ) =
∑

uv∈E(G)

[du + dv]
2

= e1[dG(2 + v2) + dG(2 + v2)]
2 + 2e1[2 + dG(2 + v2)]

2 + e1e2[(dH + 2)

+ (dH + 2)]2 + 2v2e1[(dH + 2) + dG(2 + v2)]
2

= e1[2dG(2 + v2)]
2 + 2e1[2 + dG(2 + v2)]

2 + e1e2[2(dH + 2)]2

+ 2v2e1[(dH + 2) + dG(2 + v2)]
2

HM2[ψ] ≤ e1[2∆G(2 + v2)]
2 + 2e1[2 + ∆G(2 + v2)]

2 + e1e2[(∆H + 2)2]

+ 2v2e1[∆H + 2 +∆G(2 + v2)]
2

Similarly,

HM [ψ] ≥ e1[2δG(2 + v2)]
2 + 2e1[2 + δG(2 + v2)]

2 + e1e2[(δH + 2)2]

+ 2v2e1[δH + 2 + δG(2 + v2)]
2.

Theorem 8. Let G and H be two simple connected graphs, then

SO[ψ] ≤ e1

√
2∆G

2(2 + v2)2 + 2e1

√
4 + ∆G

2(2 + v2)2 + e1e2
√

2(∆H + 2)2

+ 2v2e1

√
(∆H + 2)2 +∆G

2(2 + v2)2



Dafik et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1094-1109 1101

and

SO[ψ] ≥ e1

√
2δG

2(2 + v2)2 + 2e1

√
4 + δG

2(2 + v2)2 + e1e2
√

2(δH + 2)2

+ 2v2e1

√
(δH + 2)2 + δG

2(2 + v2)2.

Proof. Using Table1 and definition of sombor index, we have

SO[ψ] =
∑

uv∈E[G]

√
d2u + d2v

= e1

√
dG

2(2 + v2)2 + dG
2(2 + v2)2 + 2e1

√
22 + dG

2(2 + v2)2

+ e1e2
√

(dH + 2)2 + (dH + 2)2 + 2v2e1

√
(dH + 2)2 + dG

2(2 + v2)2

= e1

√
2dG

2(2 + v2)2 + 2e1

√
4 + dG

2(2 + v2)2 + e1e2
√
2(dH + 2)2

+ 2v2e1

√
(dH + 2)2 + dG

2(2 + v2)2

SO[ψ] ≤ e1

√
2∆G

2(2 + v2)2 + 2e1

√
4 + ∆G

2(2 + v2)2 + e1e2
√

2(∆H + 2)2

+ 2v2e1

√
(∆H + 2)2 +∆G

2(2 + v2)2.

Similarly,

SO[ψ] ≥ e1

√
2δG

2(2 + v2)2 + 2e1

√
4 + δG

2(2 + v2)2 + e1e2
√

2(δH + 2)2

+ 2v2e1

√
(δH + 2)2 + δG

2(2 + v2)2.

Theorem 9. Let G and H be two simple connected graphs, then

RR[ψ] ≤ e1[∆G(2+v2)]+2e1
√

2∆G(2 + v2)+e1e2[∆H +2]+2v2e1
√

(∆H + 2)∆G(2 + v2)

and

RR[ψ] ≥ e1[δG(2 + v2)] + 2e1
√

2δG(2 + v2) + e1e2[δH + 2] + 2v2e1
√

(δH + 2)δG(2 + v2).

Proof. Using Table 1 and the definition of Reciprocal Randic index, we have

RR[ψ] =
∑

uv∈E[G]

√
du.dv

= e1
√
dG(2 + v2)dG(2 + v2) + 2e1

√
2dG(2 + v2) + e1e2

√
(dH + 2)(dH + 2)

+ 2v2e1
√
(dH + 2)dG(2 + v2)

= e1[dG(2 + v2)] + 2e1
√

2dG(2 + v2) + e1e2[dH + 2] + 2v2e1
√

(dH + 2)dG(2 + v2)

RR[ψ] ≤ e1[∆G(2 + v2)] + 2e1
√

2∆G(2 + v2) + e1e2[∆H + 2] + 2v2e1
√
(∆H + 2)∆G(2 + v2)



Dafik et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1094-1109 1102

similarly,

RR[ψ] ≥ e1[δG(2 + v2)] + 2e1
√

2δG(2 + v2) + e1e2[δH + 2] + 2v2e1
√

(δH + 2)δG(2 + v2).

Theorem 10. Let G and H be two simple connected graphs, then

N [ψ] ≤ e1[
√
2∆G(2 + v2)] + 2e1

√
2 + ∆G(2 + v2) + e1e2[

√
2(∆H + 2)]

+ 2v2e1
√
(∆H + 2) + ∆G(2 + v2)

and

N [ψ] ≥ e1[
√
2δG(2 + v2)] + 2e1

√
2 + δG(2 + v2) + e1e2[

√
2(δH + 2)]

+ 2v2e1
√

(δH + 2) + δG(2 + v2).

Proof. Using table 1 and definition of Nirmala index, we have

N [ψ] =
∑

uv∈E[G]

√
du + dv

= e1
√
dG(2 + v2) + dG(2 + v2) + 2e1

√
2 + dG(2 + v2)

+ e1e2
√

(dH + 2) + (dH + 2) + 2v2e1
√
(dH + 2) + dG(2 + v2)

= e1[2dG(2 + v2)] + 2e1
√
2 + dG(2 + v2)

+ e1e2[2(dH + 2)] + 2v2e1
√

(dH + 2) + dG(2 + v2)

N [ψ] ≤ e1[
√

2∆G(2 + v2)] + 2e1
√
2 + ∆G(2 + v2) + e1e2[

√
2(∆H + 2)]

+ 2v2e1
√
(∆H + 2) + ∆G(2 + v2)

similarly,

N [ψ] ≥ e1[
√

2δG(2 + v2)] + 2e1
√

2 + δG(2 + v2) + e1e2[
√
2(δH + 2)]

+ 2v2e1
√

(δH + 2) + δG(2 + v2).

Theorem 11. Let G and H be two simple connected graphs, then

ABC[ψ] ≥ e1

[√
δG(2+v2) + δG(2 + v2)− 2

δG(2 + v2)δG(2 + v2)

]
+ 2e1

[√
2 + δG(2 + v2)− 2

2δG(2 + v2)

]

+ e1e2

[√
(δH + 2) + (δH + 2)− 2

(δH + 2) + (δH + 2)

]
+ 2v2e1

[√
(δH + 2) + δG(2 + v2)− 2

(δH + 2) + δG(2 + v2)

]
.

Proof. Using table 1 and definition of Atombond connectivity index, we have

ABC[ψ] =
∑

uv∈E[G]

√
du + dv − 2

du.dv



Dafik et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1094-1109 1103

= e1

[√
dG(2+v2) + dG(2 + v2)− 2

dG(2 + v2)dG(2 + v2)

]
+ 2e1

[√
2 + dG(2 + v2)− 2

2dG(2 + v2)

]

+ e1e2

[√
(dH + 2) + (dH + 2)− 2

(dH + 2) + (dH + 2)

]
+ 2v2e1

[√
(dH + 2) + dG(2 + v2)− 2

(dH + 2) + dG(2 + v2)

]

≥ e1

[√
δG(2+v2) + δG(2 + v2)− 2

δG(2 + v2)δG(2 + v2)

]
+ 2e1

[√
2 + δG(2 + v2)− 2

2δG(2 + v2)

]

+ e1e2

[√
(δH + 2) + (δH + 2)− 2

(δH + 2) + (δH + 2)

]
+ 2v2e1

[√
(δH + 2) + δG(2 + v2)− 2

(δH + 2) + δG(2 + v2)

]
.

Theorem 12. Let G and H be two simple connected graphs, then

ReZG2[ψ] ≥ e1

[
δ2G(2 + v2)

2

2δG(2 + v2)

]
+ 2e1

[
2δG(2 + v2)

2 + δG(2 + v2)

]
+ e1e2

[
(δH + 2)2

2(δH + 2)

]
+ 2v2e1

[
(δH + 2)δG(2 + v2)

(δH + 2) + δG(2 + v2)

]
.

Proof. Using table 1 and definition of Redefined second Zagreb index, we have

ReZG2[ψ] =
∑

uv∈E[G]

du.dv
du + dv

= e1

[
d2G(2 + v2)

2

2dG(2 + v2)

]
+ 2e1

[
2dG(2 + v2)

2 + dG(2 + v2)

]
+ e1e2

[
(dH + 2)2

2(dH + 2)

]
+ 2v2e1

[
(dH + 2)dG(2 + v2)

(dH + 2) + dG(2 + v2)

]
ReZG2 ≥ e1

[
δ2G(2 + v2)

2

2δG(2 + v2)

]
+ 2e1

[
2δG(2 + v2)

2 + δG(2 + v2)

]
+ e1e2

[
(δH + 2)2

2(δH + 2)

]
+ 2v2e1

[
(δH + 2)δG(2 + v2)

(δH + 2) + δG(2 + v2)

]
.

Observation: Inequality can change when the value of numerator of topological indices
is less than the value of denominatorof topological indices.
i.e.,

∆G ≤ degG(g), δG ≥ degG(g)



Dafik et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1094-1109 1104

Theorem 13. Let G and H be two simple connected graphs, then

GA[ψ] ≥ e1

[2√∆2
G(2 + v2)2

2∆G(2 + v2)

]
+ 2e1

[
2
√
2∆G(2 + v2)

2 + ∆G(2 + v2)

]
+ e1e2

[
2
√

(∆H + 2)2

2(∆H + 2)

]
+ 2v2e1

[
2
√

(∆H + 2)∆G(2 + v2)

(∆H + 2) + ∆G(2 + v2)

]
and

GA[ψ] ≤ e1

[2√δ2G(2 + v2)2

2δG(2 + v2)

]
+ 2e1

[
2
√
2δG(2 + v2)

2 + δG(2 + v2)

]
+ e1e2

[
2
√

(δH + 2)2

2(δH + 2)

]
+ 2v2e1

[
2
√
(δH + 2)δG(2 + v2)

(δH + 2) + δG(2 + v2)

]
.

Proof. Using Table 1 and the definition of Geometric-Arithmetic index, we have

GA[ψ] =
∑

uv∈E[G]

2
√
du.dv

du + dv

= e1

[
2
√
dG(2 + v2).dG(2 + v2)

dG(2 + v2) + dG(2 + v2)

]
+ 2e1

[
2
√

2dG(2 + v2)

2 + dG(2 + v2)

]
+ e1e2

[
2
√

(dH + 2).(dH + 2)

(dH + 2) + (dH + 2)

]
+ 2v2e1

[
2
√

(dH + 2)dG(2 + v2)

(dH + 2) + dG(2 + v2)

]

= e1

[2√d2G(2 + v2)2

2dG(2 + v2)

]
+ 2e1

[
2
√

2dG(2 + v2)

2 + dG(2 + v2)

]
+ e1e2

[
2
√
(dH + 2)2

2(dH + 2)

]
+ 2v2e1

[
2
√
(dH + 2)dG(2 + v2)

(dH + 2) + dG(2 + v2)

]

GA[ψ] ≥ e1

[2√∆2
G(2 + v2)2

2∆G(2 + v2)

]
+ 2e1

[
2
√

2∆G(2 + v2)

2 + ∆G(2 + v2)

]
+ e1e2

[
2
√

(∆H + 2)2

2(∆H + 2)

]
+ 2v2e1

[
2
√
(∆H + 2)∆G(2 + v2)

(∆H + 2) + ∆G(2 + v2)

]
Similarly,

GA[ψ] ≤ e1

[2√δ2G(2 + v2)2

2δG(2 + v2)

]
+ 2e1

[
2
√
2δG(2 + v2)

2 + δG(2 + v2)

]
+ e1e2

[
2
√

(δH + 2)2

2(δH + 2)

]
+ 2v2e1

[
2
√
(δH + 2)δG(2 + v2)

(δH + 2) + δG(2 + v2)

]
.

Theorem 14. Let G and H be two simple connected graphs, then

H[ψ] ≥ e1

[
e1

∆G(2 + v2)

]
+

[
4e1

2 + ∆G(2 + v2)

]
+

[
e1e2

(∆H + 2)

]
+

[
4v2e1

∆H + 2 +∆G(2 + v2)

]



Dafik et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1094-1109 1105

and

H[ψ] ≤ e1

[
e1

δG(2 + v2)

]
+

[
4e1

2 + δG(2 + v2)

]
+

[
e1e2

(δH + 2)

]
+

[
4v2e1

δH + 2 + δG(2 + v2)

]
.

Proof. Using table 1 and definition of Harmonic index, we have

H[ψ] =
∑

uv∈E[G]

2

du + dv

= e1

[
2

dG(2 + v2) + dG(2 + v2)

]
+ 2e1

[
2

2 + dG(2 + v2)

]
+ e1e2

[
2

(dH + 2) + (dH + 2)

]
+ 2e1v2

[
2

dH + 2 + dG(2 + v2)

]
H[ψ] ≥ e1

[
e1

∆G(2 + v2)

]
+

[
4e1

2 + ∆G(2 + v2)

]
+

[
e1e2

(∆H + 2)

]
+

[
4v2e1

∆H + 2 +∆G(2 + v2)

]
Similarly,

H[ψ] ≤ e1

[
e1

δG(2 + v2)

]
+

[
4e1

2 + δG(2 + v2)

]
+

[
e1e2

(δH + 2)

]
+

[
4v2e1

δH + 2 + δG(2 + v2)

]
.

Theorem 15. Let G and H be two simple connected graphs, then

SC[ψ] ≥
[

2e1√
2∆G(2 + v2)

]
+

[
4e1√

2 + ∆G(2 + v2)

]
+

[
2e1e2√

2(∆H + 2)

]
+

[
4v2e1√

(∆H + 2) + ∆G(2 + v2)

]
and

SC[ψ] ≤
[

2e1√
2δG(2 + v2)

]
+

[
4e1√

2 + δG(2 + v2)

]
+

[
2e1e2√
2(δH + 2)

]
+

[
4v2e1√

(δH + 2) + δG(2 + v2)

]
.

Proof. Using Table 1 and the definition of Sum-connectivity index, we have

SC[ψ] =
∑

uv∈E[G]

2√
du + dv

= e1

[
2√

dG(2 + v2) + dG(2 + v2)

]
+ 2e1

[
2√

2 + dG(2 + v2)

]



Dafik et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1094-1109 1106

+ e1e2

[
2√

(dH + 2) + (dH + 2)

]
+ 2e1v2

[
2√

dH + 2 + dG(2 + v2)

]
Sc[ψ] ≥

[
2e1√

2∆G(2 + v2)

]
+

[
4e1√

2 + ∆G(2 + v2)

]
+

[
2e1e2√

2(∆H + 2)

]
+

[
4v2e1√

(∆H + 2) + ∆G(2 + v2)

]
Similarly,

SC[ψ] ≤
[

2e1√
2δG(2 + v2)

]
+

[
4e1√

2 + δG(2 + v2)

]
+

[
2e1e2√
2(δH + 2)

]
+

[
4v2e1√

(δH + 2) + δG(2 + v2)

]
.

Theorem 16. Let G and H be two simple connected graphs, then

ReZG1[ψ] ≥ e1

[
2e1

∆G(2 + v2)

]
+ e1

[
2 + ∆G(2 + v2)

∆G(2 + v2)

]
+

[
2e1e2

(∆H + 2)

]
+ 2v2e1

[
(∆H + 2)∆G(2 + v2)

(∆H + 2)∆G(2 + v2)

]
.

and

ReZG1[ψ] ≤ e1

[
2e1

δG(2 + v2)

]
+ e1

[
2 + δG(2 + v2)

δG(2 + v2)

]
+

[
2e1e2

(δH + 2)

]
+ 2v2e1

[
(δH + 2)δG(2 + v2)

(δH + 2)δG(2 + v2)

]
.

Proof. Using Table 1 and the definition of Redefined first Zagreb index, we have

ReZG1[ψ] =
∑

uv∈E[G]

[
du + dv
du.dv

]

= e1

[
2dG(2 + v2)

d2G(2 + v2)2

]
+ 2e1

[
2 + dG(2 + v2)

2dG(2 + v2)

]
+ e1e2

[
2(dH + 2)

(dH + 2)2

]
+ 2v2e1

[
(dH + 2) + dG(2 + v2)

(dH + 2)dG(2 + v2)

]
ReZG1[ψ] ≥ e1

[
2e1

∆G(2 + v2)

]
+ e1

[
2 + ∆G(2 + v2)

∆G(2 + v2)

]
+

[
2e1e2

(∆H + 2)

]
+ 2v2e1

[
(∆H + 2)∆G(2 + v2)

(∆H + 2)∆G(2 + v2)

]
.

Similarly,

ReZG1[ψ] ≤ e1

[
2e1

δG(2 + v2)

]
+ e1

[
2 + δG(2 + v2)

δG(2 + v2)

]
+

[
2e1e2

(δH + 2)

]



REFERENCES 1107

+ 2v2e1

[
(δH + 2)δG(2 + v2)

(δH + 2)δG(2 + v2)

]
.

3. Conclusion

In this work, we have considered ψ graph and concentrated a few important topological
indices and determine their bounds. In the same way, we can examine the different classes
of topological indices and determine their corresponding bounds for ψ graph.

Acknowledgements

The author expresses deepest gratitude to the the Department of backward classes
welfare (BCWD) Karnataka Government, and the Department of Mathematics, V. S. K.
University, Ballari, India, as well as PUI-PT Combinatorics and Graph, CGANT Uni-
versity of Jember, Indonesia, from the support on finishing this paper. I also extend
the acknowledgement to the anonymous referee for their valuable comments and fruitful
suggestions which enhanced the readability of the paper.

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