EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 5, 2020, 1260-1269
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Special Issue Dedicated to
Professor Hari M. Srivastava

On the Occasion of his 80th Birthday

Harmonic Index and Zagreb Indices of Vertex-Semitotal
Graphs

Aysun Yurttas Gunes1, Muge Togan1, Musa Demirci1,∗, Ismail Naci Cangul1

1 Department of Mathematics, Bursa Uludag University, 16059 Bursa, Turkey

Abstract. Graph theory is one of the rising areas in mathematics due to its applications in
many areas of science. Amongst several study areas in graph theory, spectral graph theory and
topological descriptors are in front rows. These descriptors are widely used in QSPR/QSAR studies
in mathematical chemistry. Vertex-semitotal graphs are one of the derived graph classes which
are useful in calculating several physico-chemical properties of molecular structures by means of
molecular graphs modelling the molecules. In this paper, several topological descriptors of vertex-
semitotal graphs are calculated. Some new relations on these values are obtained by means of a
recently defined graph invariant called omega invariant.

1. Introduction

Several topological graph indices have been defined and studied by many mathemati-
cians and chemists. They are defined as topological graph invariants measuring several
physical, chemical, pharmacological, pharmaceutical, biological, etc. properties of graphs
which are modelling real life situations. They can be grouped mainly into three classes
according to the way they are defined: by vertex degrees, by matrices or by distances. We
consider degree based-topological indices of some derived graphs through this paper.

Let G = (V,E) be a simple graph with | V (G) |= n vertices and | E(G) |= m edges,
where V (G) = {v1, v2, · · · , vn} and E(G) = {vivj : vi, vj ∈ V (G)}. That is, we do not

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i5.3725

Email addresses: ayurttas@uludag.edu.tr (A. Yurttas Gunes), capkinm@uludag.edu.tr (M. Togan),
mdemirci@uludag.edu.tr (M. Demirci), cangul@uludag.edu.tr (I. N. Cangul)

https://www.ejpam.com 1260 c© 2020 EJPAM All rights reserved.



M. Demirci et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1260-1269 1261

allow loops or multiple edges. For any vertex v ∈ V (G), we denote the degree of v by
dG(v) or dv.

If vi and vj are adjacent vertices of G, and if the edge e connects them, this situation
will be denoted by e = vivj . In such a case, the vertices vi and vj are called adjacent
vertices and the edge e is said to be incident to vi and vj . Adjacency and incidency play
a very important role in the spectral graph theory, the sub area of graph theory dealing
with linear algebraic study of graphs. The smallest and biggest vertex degrees in a graph
will be denoted by δ and ∆, respectively.

Written with multiplicities, a degree sequence in general is written as

D = {1(a1), 2(a2), 3(a3), · · · ,∆(a∆)}.

Let D be a set of some non-decreasing non-negative integers. We say that a graph G
is a realization of the set D if the degree sequence of G is equal to D.

Definition 1. [7] Let D = {1(a1), 2(a2), 3(a3), · · · ,∆(a∆)} be a realizable degree sequence
and G be one of its realizations. The Ω(G) of G is defined in terms of the degree sequence
as

Ω(G) = a3 + 2a4 + 3a5 + · · ·+ (∆− 2)a∆ − a1

=
∆∑
i=1

ai(i− 2).

A vertex-semitotal graph T1(G) is constructed fromG by inserting a new vertex for each
edge of G and then by joining every inserted vertex to the end vertices of the corresponding
edge, that is, by replacing each edge by a triangle. See Fig. 1. Thus |V (T1)| = |V (G)|+
|E(G)| = n+m and |E(T1)| = |E(S)|+ |E(G)| = 2m+m = 3m.

Two of the most important topological graph indices are called the first and second
Zagreb indices denoted by M1(G) and M2(G), respectively:

M1(G) =
∑

u∈V (G)

d2
G(u) and M2(G) =

∑
uv∈E(G)

dG(u)dG(v). (1)

They were first defined in 1972 by Gutman and Trinajstic, [12], and are referred to due
to their uses in QSAR and QSPR studies. In [4], some results on the first Zagreb index
together with some other indices are given. For some graph operations, these indices are
calculated in [5, 14, 17].

The F -index or forgotten index of a graph G denoted by F (G) or M3(G) is defined as

F (G) =
∑

u∈V (G)

d3
G(u). (2)



M. Demirci et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1260-1269 1262

Figure 1 The vertex-semitotal graph T1(G) of T3,2

It was first appeared in the study of structure-dependency of total π-electron energy in
1972, [12]. Recently, this sum was named as the forgotten index or the F -index by Furtula
and Gutman, [10].

The hyper-Zagreb index was defined as a variety of the classical Zagreb indices as

HM(G) =
∑
uv∈E

(du + dv)
2, (3)

see e.g. [10].

Inspired by the study of heat of formation for heptanes and octanes, in [9] Furtula
et al. proposed an index called Augmented Zagreb index which gives a better prediction
power. It is defined by

AZI(G) =
∑

uv∈E(G)

(
dudv

du + dv − 2

)3

. (4)

The Harmonic index was introduced by Zhong [19] who found that it correlates well
with Π-electron energy of benzenoid hydrocarbons and defined as

H(G) =
∑

uv∈E(G)

2

du + dv
. (5)

Ranjini et al., [16], introduced the re-defined Zagreb indices, i.e. the redefined first,



M. Demirci et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1260-1269 1263

second and third Zagreb indices for a graph G and these are defined as

ReZG1(G) =
∑

uv∈E(G)

du + dv
du · dv

, (6)

ReZG2(G) =
∑

uv∈E(G)

du · dv
du + dv

, (7)

ReZG3(G) =
∑

uv∈E(G)

(du · dv)(du + dv). (8)

Milicevic et al., [15], reformulated the Zagreb indices in terms of the edge degrees
instead of the vertex-degrees as

EM1(G) =
∑

e=uv∈E(G)

d(e)2 and EM2(G) =
∑

e f∈E(G)

dG(e)dG(f). (9)

Aram and Dehgardi, [1], introduced the concept of reformulated F-index as

RF (G) =
∑

e=uv∈E(G)

d(e)3. (10)

Eliasi et. al. [8] introduced the multiplicative sum Zagreb index of G which is denoted by∏∗
1(G) and defined by ∏∗

1
(G) =

∏
uv∈E(G)

(dG(u) + dG(v)). (11)

Xu et. al. [18] introduced the total multiplicative sum Zagreb index of a graph G denoted
by
∏T (G) and defined by∏T

(G) =
∏

u,v∈V (G)

(dG(u) + dG(v)). (12)

Topological indices of some derived graphs, as subdivision, total, semitotal, line, par-
aline graphs are studied in [2] and [13]. In this paper, we examine some degree-based
topological indices of vertex-semitotal graph which also is one of the derived graphs, and
find relations between these topological indices.

2. Main Results

We first recall some results on the topological indices of the vertex-semitotal graphs:

Proposition 1. [11] Let T1(G) be the vertex-semitotal graph of the graph G of order
n = n(G) and size m = m(G). Then

M1(T1(G)) = 4M1(G) + 4m(G).



M. Demirci et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1260-1269 1264

Theorem 1. [2] Let G be a graph of order n = n(G) and size m = m(G). Then

M2(T1(G)) = 2EM1(G) + EM2(G) + 2M1(G) +M2(G) + F (G)− 4m(G).

Proposition 2. [6] Let G be a graph of order n = n(G) and size m = m(G). Then

F (T1(G)) = 8F (G) + 8m(G).

Theorem 2. [13] If T1(G) is a vertex-semitotal graph of G of order n = n(G) and size
m = m(G). Then

EM1(T1(G)) = 8(F (G)−M1(G) +M2(G)) + 4m(G)

and

EM2(T1(G)) =
1

3

(
14(4M1(G)) + 4EF (G) + 68m(G)

)
+ 4EM2(G) + 6F (G)− 30M1(G) + 28M2(G)

where αM1(G) =
∑

v∈V (G) d(v)α and EF (G) is the reformulated forgotten index.

Theorem 3. [3] Let G be a graph of order n = n(G) and size m = m(G). Then∏
1
(T1(G)) =

∏
1
(G)

[∏∗

1
(G)
]2
.

and ∏
2
(T1(G)) =

∏
2
(G)
∏∗

2
(G).

Now we will determine some well-known Zagreb indices of vertex-semitotal graph of
G.

Lemma 1. Let G be a connected simple graph of order n = n(G) and size m = m(G) and
let T1(G) be the vertex-semitotal graph of G. Then,

Ω(T1(G))− Ω(G) = 2m(G).

The proof is clear from the definition of Ω invariant of G.

Theorem 4. Let G be a graph with order n = n(G) and size m = m(G). Then the hyper
Zagreb index of T1(G) is

HM(T1(G)) = 4 (2M1(G) + F (G) +HM(G) + 2m(G)) .

Proof. By Eqn. (3), we have

HM(T1(G)) =
∑

vivj∈E(T1(G))

[
dT1(G)(vi) + dT1(G)(vj)

]2



M. Demirci et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1260-1269 1265

=
∑

vivj∈E(G)

[
dT1(G)(vi) + dT1(G)(vj)

]2
+

∑
vicij∈E(T1(G))

[
dT1(G)(vi) + 2

]2
= 4

∑
vivj∈E(G)

(dG(vi) + dG(vj))
2 + 4

∑
vi∈V (G)

(1 + dG(vi))
2dG(vi),

and the result follows.

Theorem 5. Let G be a graph with order n = n(G) and size m = m(G). Then augmented
Zagreb index of T1(G) is

AZI(T1(G)) = 8
∑

vivj∈E(G)

(
dG(vi)dG(vj)

dG(vi) + dG(vj)− 1

)3

+ 16m(G).

Proof. Using Eqn.(4), we get

AZI(T1(G)) =
∑

vivj∈E(G)

(
dT1(G)(vi) · dT1(G)(vj)

dT1(G)(vi) + dT1(G)(vj)− 2

)3

+
∑

vicij∈E(T1(G))

(
2.2dG(vi)

2 + 2dG(vi)− 2

)3

dG(vi)

=
∑

vivj∈E(G)

(
2dG(vi) · 2dG(vj)

2dG(vi) + 2dG(vj)− 2

)3

+ 8
∑

vi∈V (G)

dG(vi),

and the result follows.

Theorem 6. Let G be a graph with order n = n(G) and size m = m(G). Re-defined
versions of Zagreb indices of T1(G) are

i) ReZG1(T1(G)) = 1
2 (ReZG1(G) + n(G)) +m(G).

ii) ReZG2(T1(G)) = 2
[
ReZG2(G) +

∑
u∈V (G)

d2
G(vi)

1+dG(vi)

]
.

iii) ReZG3(T1(G)) = ReZG3(G) + 8 (M1(G) + F (G)) +m(G).

Proof. From Eqn. (6), we have

ReZG1(T1(G)) =
∑

vivj∈E(T1(G))

dT1(G)(vi) + dT1(G)(vj)

dT1(G)(vi) · dT1(G)(vj)

=
∑

vivj∈E(G)

2dG(vi) + 2dG(vj)

2dG(vi) · 2dG(vj)
+

∑
vicij∈E(T1(G))

2 + 2dG(vi)

2 · 2dG(vi)
· dG(vi)



M. Demirci et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1260-1269 1266

=
1

2

∑
vivj∈E(G)

dG(vi) + dG(vj)

dG(vi) · dG(vj)
+

1

2

∑
vi∈V (G)

(1 + dG(vi)),

and the result follows.

Using Eqns.(7) and (8), we get the results for (ii) and (iii) by similar methods.

Theorem 7. Let G be a graph with order n = n(G) and size m = m(G). Reformulated
forgotten index of T1(G) is

RF (T1(G)) = 8
(
2M4(G) + 3ReZG3(G)−m(G)

)
− 24 (F (G) + 2M2(G)−M1(G)) .

Proof. For vertex-semitotal graph T1(G) of a graph G, there are two types of vertices:
Firstly, the vertices corresponding to the vertices of G, secondly, the vertices corresponding
to the edges of G. We will denote them vi and cij , respectively. Depending on the nature
of end vertices, we can divide the edges of T1 into two types:

i) vivj-edge: an edge between two vertices in G.
ii) vicij-edge: an edge between the vertices of G and the vertices corresponding to the
edges of G.
Using Eqn. (9), we have

RF (T1(G)) =
∑

e∈E(T1(G))

(de)
3

=
∑

evivj∈E(T1)

dT1(evivj )
3 +

∑
evicij∈E(T1)

dT1(evicij )
3

=
∑

vivj∈E(T1)

[dT1(vi) + dT1(vj)− 2]3 +
∑

vicij∈E(T1)

[dT1(vi) + dT1(cij)− 2]3.

For vicij-edges in the second term, it is clear that every vi vertex of T1(G) is connected
with dG(vi) cij vertices, each of degree 2. Therefore, corresponding to every vertex vi in
G, there are dG(vi) edges in T1 each of edge degree [2dG(vi) + 2− 2]. So,

RF (T1(G)) =
∑

vivj∈E(G)

[2dG(vi) + 2dG(vj)− 2]3 +
∑

vi∈V (G)

dG(vi)[2dG(vi) + 2− 2]3

= 8

 ∑
vivj∈E(G)

(d3
G(vi) + d3

G(vj)) + 3
∑

vivj∈E(G)

dG(vi)dG(vj)(dG(vi) + dG(vj))


− 6

∑
vivj∈E(G)

(2dG(vi) + dG(vj))
2 + 12

∑
vivj∈E(G)

(2dG(vi) + dG(vj))

−
∑

vivj∈E(G)

8 + 8
∑

vi∈V (G)

d4
G(vi)

= 8
[
M4(G) + 3ReZG3(G)−m(G)

]
− 24 [F (G) + 2M2(G)−M1(G)] .



M. Demirci et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1260-1269 1267

Theorem 8. Let G be a graph of order n = n(G) and size m = m(G). Multiplicative sum
Zagreb index of T1(G) is∏∗

1
(T1(G)) = 4

∏∗

1
(G)

∏
vi∈V (G)

(1 + dG(vi)).

Proof.∏∗

1
(T1(G)) =

∏
vivj∈E(T (G))

(dT (G)(vi) + dT (G)(vj))

=
∏

vivj∈E(G)

(dT (G)(vi) + dT (G)(vj)) ·
∏

vicij∈E(T (G))

(2 + dT (G)(vi))

=
∏

vivj∈E(G)

(2dG(vi) + 2dG(vj)) · 2
∏

vicij∈E(T (G))

(1 + dG(vi))

and the result follows.

Theorem 9. Let G be a graph of order n = n(G) and size m = m(G). Total multiplicative
sum Zagreb index of T1(G) is∏T

(T1(G)) = 2m(G)2+1
∏T

(G)
∏

vi∈V (G)

(1 + dG(vi))
m(G).

Proof. From the definition of
∏T (G), we have∏T

(T1(G)) =
∏

vi,vj∈V (T (G))

(dT (G)(vi) + dT (G)(vj))

=
∏

vi,vj∈V (G)

(2d(G)(vi) + 2d(G)(vj)) ·
∏

vi,cij∈V (T (G))

(2 + 2d(G)(vi))
m(G)

·
∏

cij ,ckt∈V (S(G))

(2 + d(G)(vi))

= 2
∏T

(G) · 2m
∏

vi∈V (G)

(1 + dG(vi))
m(G) · 4(m(G)

2 )

and the result follows.

3. Conclusions

In this paper, we obtained the formulae for the topological indices, especially the
Zagreb indices and harmonic index, of some class of derived graphs called vertex-semitotal
graphs. We used for the first time a newly introduced graph invariant called omega
invariant to obtain some of the relations. The methods used here can be applied to all
topological graph indices and to other derived graphs and graph operations.



REFERENCES 1268

References

[1] H Aram and N Dehgardi. Reformulated F-index of graph operations. Commun.
Comb. Optim., 2:87–98, 2017.

[2] B Basavanagoud, I Gutman, and C S Gali. On Second Zagreb Index and Coindex of
Some Derived Graphs. Kragujevac J. Sci., 37:113–121, 2015.

[3] B Basavanagoud and S Patil. Multiplicative Zagreb Indices and Coindices of Some
Derived Graphs. Opuscula Math., 36(3):287–299, 2016.

[4] K C Das, N Akgunes, M Togan, A Yurttas, I N Cangul, and A S Cevik. On the first
Zagreb index and multiplicative Zagreb coindices of graphs. Analele Stiintifice ale
Universitatii Ovidius Constanta, 24(1):153–176, 2016.

[5] K C Das, A Yurttas, M Togan, I N Cangul, and A S Cevik. The multiplicative Zagreb
indices of graph operations. Journal of Inequalities and Applications, 90, 2013.

[6] N De. F-Index and Coindex of Some Derived Graphs. Bulletin of the International
Mathematical Virtual Institute, 8:81–88, 2018.

[7] S Delen and I N Cangul. A New Graph Invariant. Turkish Journal of Analysis and
Number Theory, 6(1):30–33, 2018.

[8] M Eliasi, A Iranmanesh, and I Gutman. Multiplicative Versions of First Zagreb Index.
MATCH Commun. Math. Comput. Chem., 68:217–230, 2012.

[9] B Furtula, A Graovac, and D Vukicevic. Augmented Zagreb Index. Journal of
Mathematical Chemistry, 48:370–380, 2010.

[10] B Furtula and I Gutman. A Forgotten Topological Index. J. Math. Chem., 53(4):1184–
1190, 2015.

[11] I Gutman, B Furtula, Z K Vukicevic, and G Popidova. On Zagreb Indices and
Coindices. MATCH Commun. Math. Comput. Chem., 74:5–16, 2015.

[12] I Gutman and N Trinajstic. Graph theory and molecular orbitals III. Total π-electron
energy of alternant hydrocarbons. Chem. Phys. Lett., 17(1):535–538, 1972.

[13] J B Liu, A Bahadur, M A Malik, H M A Siddiqui, and M Imran. Reformulated
Zagreb Indices of Some Derived Graphs. Mathematics, 7(4):366, 2019.

[14] V Lokesha, R Shruti, P S Ranjini and A. S. Cevik. On Certain Topological Indices of
Nanostructures Using Q(G) and R(G) Operators. Communications Fac. Sci. Univ.
Ank. Ser. A1: Math. and Stat., 66(2):1-10, 2018.

[15] A Milicevic, S Nikolic, and N Trinajstic. On reformulated Zagreb indices. Mol.
Divers., 8:393–399, 2004.



REFERENCES 1269

[16] P S Ranjini, V Lokesha, and A Usha. Relation between phenylene and hexagonal
squeeze using harmonic index. Int J Graph Theory, 1:116–121, 2013.

[17] B S Shetty, V Lokesha, and P S Ranjini. On the Harmonic index of graph operations.
Transactions on Combinatorics, 4(4):5-14, 2015.

[18] K Xu, K C Das, and K Tang. On the Multiplicative Zagreb Coindex of Graphs.
Opuscula Math., 33:191–204, 2013.

[19] L Zhong. The harmonic index on graphs. Applied Mathematics Letters, 25:561–566,
2012.