EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 3, 2023, 1794-1808
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On weighted vertex and edge Mostar index for trees
and cacti with fixed parameter

Farwa Asmat1,∗, Humaira Asmat2, Sameh Askar3, Hijaz Ahamd4,6,7,
Muhammad Ijaz Khan5

1,5 School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China
2 Department of Mathematics and Statistics, University of Lahore, Pakistan
3 Department of Statistics and Operations Research, College of Science,
King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
4 Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio
Emanuele-II, 39,00186 Roma, Italy
5 Department of Mechanical Engineering, Lebanese American University, Beirut, Lebanon
6 Near East University, Operational Research Center in Healthcare, Nicosia 99138, TRNC
Mersin 10, Turkey
7 Department of Computer Science and Mathematics, Lebanese American University, Beirut,
Lebanon

Abstract. It was introduced by Došlić and Ivica et al. (Journal of Mathematical chemistry, 56(10)
(2018): 2995–3013), as an innovative graph-theoretic topological identifier, the Mostar index is
significant in simulating compounds thermodynamic properties in simulations, which is defined as
sum of absolute values of the differences among nu(e|Ω) and nv(e|Ω) over all lines e = uv ∈ Ω,
where nu(e|Ω) (resp. nv(e|Ω)) is the collection of vertices of Ω closer to vertex u (resp. v) than to
vertex v (resp. u). Let C(n, k) be the set of all n-vertex cactus graphs with exactly k cycles and
T (n, d) be the set of all n-vertex tree graphs with diameter d.
It is said that a cactus is a connected graph with blocks that comprise of either cycles or edges.
Beginning with the weighted Mostar index of graphs, we developed certain transformations that
either increase or decrease the index. To advance this analysis, we determine the extreme graphs
where the maximum and minimum values of the weighted edge Mostar index are accomplished.
Moreover, we compute the maximum weighted vertex Mostar invariant for trees with order n and
fixed diameter d.

2020 Mathematics Subject Classifications: 05C05, 05C35, 05C12, 05C92

Key Words and Phrases: Weighted Mostar index, Trees, Diameter, Extremal values, Cactus
graphs

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4722

Email addresses: farwaasmat@pku.edu.cn (F. Asmat), humairaasmat17@gmail.com (H. Asmat),
saskar@ksu.edu.sa (S. Askar), ahmad.hijaz@uninettuno.it (H. Ahmad),
hmikhan@math.qau.edu.pk (M. I. Khan)

https://www.ejpam.com 1794 © 2023 EJPAM All rights reserved.



F. Asmat et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1794-1808 1795

1. Introduction

Graph theory is a subfield of mathematics that studies the properties of graphs. It is
currently being applied in a variety of scientific disciplines including chemistry, engineer-
ing, and physics. Graph theory can be used to model and study various problems and
phenomena in mathematics, computer science, biology, sociology, and operations research.
One of the tools that graph theory provides is the concept of topological descriptors, which
are numerical values that capture some aspects of the structure and properties of graphs.

Quantitative structure-property/ quantitative structure-activity relationship (QSPR/QSAR)
schemes are utilized as regression models in reticular chemistry to correlate with various
of biological and physicochemical activities. Harold Wiener [31] pioneered a novel way
to estimate the boiling point of alkanes from their molecular shape. He converted the
chemical structure into a single number that reflected its characteristics. This number,
called a topological descriptor or index, was obtained from a chemical graph, a simplified
representation of an organic molecule as a network of atoms and bonds. This technique
also enables us to model many other properties of molecules, such as their behavior under
extreme conditions, their energy release or absorption, and their interaction with living
systems [4, 35]. In this study, we apply topological descriptors to develop robust regression
models for these properties.

Topological descriptors play a crucial role in the definitional work done in the chemical
sciences, mathematical chemistry, chemical graph theory, and pharmaceutical science; for
example, topological indices for bond connectivity are used to quantify properties like
branching, compactness, centrality, regularity, variability, bioactivity, etc [27]. There are
remarkable subclasses of indices known as vertex and edge bond-additive indices or degree
based indices that attempt to capture some valuable aspects of complete graphs by adding
up the contributions of individual vertices and/or edges. It garnered a lot of interest in the
context of complex networks and in more traditional chemical graph theory applications.
A distance based index is a invariant based on the distance between the vertices or edges
of any graph [20].

Wiener [32] introduced the first graph index based on distance between vertices of a
graph, which is described as

W (Ω) =
∑

u,v∈V (Ω)

dΩ(u, v),

where dΩ(u, v) represents the shortest distance between u and v. Many researchers have
been exclusively studied the Wiener index and mostly characterize the sharp lower and up-
per bounds for different graph families which can be read in [5, 8, 10, 33] and then research
extend to discover the new form of Wiener index named polarity index WP (Ω), which is
restricted to distance 3 among all unordered pairs of vertices in Ω. The applications and
detailed classification of Wiener polarity index have been discussed in [18, 22].

Consider an edge e = uv ∈ Ω and the following three sets defined as Nu(e) = {k ∈
VΩ : d(u, k) < d(v, k)}, Nv(e) = {k ∈ VΩ : d(v, k) < d(u, k)}, No(e) = {k ∈ VΩ : d(u, k) =
d(v, k)}. Thus, the divisions of nodes in Ω with respect to edge e are represented by



F. Asmat et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1794-1808 1796

[Nu(e), Nv(e), No(e)]. The collection of vertices in Nu(e), Nv(e) and No(e) are symbolized
by nu(e|Ω), nv(e|Ω), and no(e|Ω), respectively. The mathematical term of Wiener index
is explained as follows [3],

We(Ω) =
∑

uv∈E(Ω)

nu(e|Ω)nv(e|Ω),

where all right-hand side sums have n − 1, and their individual values can be roughly
estimated. Mostar index was recently introduced by Doslić et al. (Journal of Mathematical
chemistry, 56(10) (2018): 2995–3013), which is defined as

Mov(Ω) =
∑

uv∈E(Ω)

|nu(e|Ω)− nv(e|Ω)|,

where nu(e|Ω) (resp. nv(e|Ω)) is the collection of vertices of Ω closer to vertex u (resp. v)
than to vertex v (resp. u).

Arockiaraj and Clement, et al. (SAR and QSAR in Environmental Research, 31(3)
(2020): 187–208) have recently proposed two new topological indices in this vein, the
weighted edge Mostar index and the weighted vertex Mostar index, which are defined as

Mowv (Ω) =
∑

uv∈E(Ω)

(dΩ(u) + dΩ(v))|nu(e|Ω)− nv(e|Ω)|,

Mowe (Ω) =
∑

uv∈E(Ω)

(dΩ(u) + dΩ(v))|mu(e|Ω)−mv(e|Ω)|,

where mu(e|Ω) (resp. mv(e|Ω)) is the collection of edges of Ω closer to vertex u (resp. v)
than to vertex v (resp. u). Many researchers have been extensively worked on different
distance based indices, see [1, 6, 11, 16, 17, 36]. Motivated by the success of previous
research, Došlić and Ivica et al. recently introduced Mostar invariant (Journal of Mathe-
matical chemistry, 56(10) (2018): 2995–3013), which belongs to bond-additive indices as
they capture the relevant properties of a graph. Many research works have been done on
Mostar index, read [13, 14, 23, 25].

A graph is said to be cactus if all of its vertices must be either edges or cycles, and no
two cycles can share more than one vertex. Readers can read many works on cactus graph
herein [9, 21, 28–30]. Extremal bicyclic graphs were obtained by Tepeh with respect to
the Mostar index [26], while extremal catacondensed benzenoids were established by Deng
[12]. There are a number of different conclusions and points of view that can be drawn
from the Mostar index, which are all explained by Ali [2]. Imran et al., [19] computed
Weighted Mostar Invariants of Phthalocyanines, Triazine-Based and Nanostar Dendrimers.
Brezovnik [7] studied Szeged and Mostar root-indices of graphs.

Consider C(n, k), the collection of all cactus graphs with n vertices and k cycles.
Inspired by the existing literature on cactus graphs and weighted edge Mostar index, we
extend the previous results to a more general setting by using the extremal graphs as an
illustrative example. We then establish an upper bound for the weighted edge Mostar



F. Asmat et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1794-1808 1797

index and identify the extremal graph among all the graphs in C(n, k). Let T (n, d) denote
the set of tree graphs with n vertices and diameter d. Let T (̂n, d) ∈ T (n, d) be the special
tree graph with diameter d and n − d − 1 pendent vertices attached to a fixed vertex.
Finally, we determine the maximum value of the weighted vertex Mostar index over all
the tree graphs.

2. Preliminaries results and Notations

Here, we only deal with simple finite connected graphs. Other notations can be studied
in [15]. Let Ω be a connected graph comprises vertex set V (Ω) and edge set E(Ω). For
an edge uv ∈ E(Ω), the graph Ω − uv is obtained by removing uv ∈ E(Ω) from Ω. For
any vertex u ∈ V (Ω), let Nu(Ω) represents the number of edges incidents to u in Ω and
dΩ(u) = |Nu(Ω)| represents the degree of u. A vertex with exactly one degree is called
pendant. A cut vertex of a graph is any vertex that when it removed the number of
connected components of this graph increases. Similarly, An edge is called cut edge if, by
deleting that edge, the graph is converted into exactly two components. Consider that Pn,
Sn, Cn and Kn the path, star, cycle, and complete graph with n vertices, respectively.

Let Sn be the star of order n. Denote by S∗n the graph generated by associating one
new edge among the leaves of the star Sn. Let S∗,kn be the generated graph of order n
constructed by associating k new edges between the leaves in a star Sn. In particular, S∗n
is just S∗,1n .

Theorem 1. [34] Let Ω ∈ C(n, k) be a connected graph. Then

(i) for n ≥ 10 and n < 4k then Moe(Ω) ≤ 2n2 − 8n + (24 − 4n)k with equality if and
only if Ω ∼= Ωn(3, 3, 3..., 3︸ ︷︷ ︸

4k−n

, 4, 4, 4, ..., 4︸ ︷︷ ︸
n−3k

).

(ii) for n ≥ 10 and n ≥ 4k then Moe(Ω) ≤ n2 − n − 12k with equality if and only if
Ω ∼= Ωn(4, 4, 4, ..., 4).

(iii) for n = 9, then Moe(Ω) ≤ 72− 12k with equality if and only if Ω ∼= Ω9.

(iv) for n ≤ 9, then Moe(Ω) ≤ n2 − n − (n + 3)k with equality if and only if Ω ∼=
Ωn(3, 3, 3, ..., 3).

The second maximum edge Mostar index for C(n, k) with the following given conditions
determined by Liu et al [24].

Theorem 2. [34] Let Ĉ(n, k) ∈ C(n, k) be a connected graph achieving maximum edge
Mostar index for n ≥ 3k + 2, k ≥ 2 and n ≥ 9, k = 1. For any Ω ∈ C(n, k), we have

Moe(Ω) < Moe(Ĉ(n, k)).

Lemma 1. [2] Suppose that Ω is a connected graph and K is an induced subgraph of Ω
such that K is a tree and connected with Ω by cut vertex u. Consider that K transform



F. Asmat et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1794-1808 1798

to star graph centered at u, then Weighted vertex index Mowv (Ω) increases (unless K is
already such a star). Similarly, if K transform to path graph with end vertex u, then
weighted vertex index Mowv (Ω) decreases.

3. Main results

In this section, we present our main results of this paper. More precisely, we have the
following two results.

Theorem 3. Among all the tree graphs in T (n, d) the T (n̂, d), for n ≥ d + 1 and d ≥ 6
has maximum weighted vertex Mostar index. Thus for any Ω ∈ T (n, d), we have

Mowv (Ω) < Mowv (T (n̂, d))

We have following result by using Theorem 3.

Corollary 1. Let Ω ∈ T (n, d) be the tree graph with n ≥ 2 and d ≥ 2, then we have the
following.

Mowv (Ω) =


n3 − 3n2 + 2n n ≥ d+ 1, d = 2

n3 − 5n2 + 10n− 12 n ≥ d+ 1, d = 3

n3 − (2m+ 1)n2 + d2 + 5d− 4− 12(n− 12) n ≥ d+ 1, m ≥ 1, and d ≥ 6

(1)

Theorem 4. For any graph Ω ∈ C(n, k) where n ≥ 2k + 1,

Mowe (Ω) ≤ n3 + (k − 4)n2 + (−3k + 5)n− (2n+ 4), ∀n ≥ 1

with the equality holds if and only if Ω ∼= S∗,kn .

4. Proof of main results

First of all, some basic lemmas are proved so that the main result can be proved easily.
In Lemma 2, we establish a graph Ω2 by converting a cut edge uv into a pendent edge uv
in Ω1, such that the new graph Ω2 has a greater weighted vertex Mostar invariant.

Lemma 2. Suppose that two subgraphs T1 and T2 such that connected by an edge uv,
where u ∈ V (T1) and v ∈ V (T2), and acquired the graph Ω1. Now, we construct the new
graph Ω2 by deleting the cut edge uv and associating a pendent edges at central vertex in
Ω1. Then Mowv (Ω1) < Mowv (Ω2).

Proof. Suppose T1 and T2 be subgraphs of Ω1. By construction of Ω2, the number of
closer vertices of end vertices of the fixed edge of T1 and T2 in Ω1 remains same in Ω2,
respectively. Therefore, for an edge xy ∈ E(Tm) for m ∈ {1, 2}, we have nx(e|Ω1)(x) =
nx(e|Ω2)(x) and ny(e|Ω1)(y) = ny(e|Ω2)(y).



F. Asmat et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1794-1808 1799

For the cut edge uv in Ω1, we have nu(e|Ω1)(u) = nv(e|Ω1)(v) = |E(T1)|+1. Similarly
for graph Ω2, nu(e|Ω2)(u) = |E(T1)|+ |E(T2|+ 1 and nv(e|Ω2)(v) = 1.

By using the definition of weighted vertex Mostar index, we have

Mowv (Ω) =
∑

e=uv∈E(Ω)

(dΩ(u) + dΩ(v))|nu(e|Ω)(v)− nv(e|Ω)(v)|,

Mowv (Ω1)−Mowv (Ω2) = (dΩ(u) + dΩ(v))|nu(e|Ω1)(u)− nv(e|Ω1)(u)|

+
2∑

m=1

∑
xy∈E(T1)

(dΩ(x) + dΩ(y))|nx(e|Ω1)(x)− ny(e|Ω1)(y)|

− (dΩ(u) + dΩ(v))|nu(e|Ω2)(u)− nv(e|Ω2)(v)|

−
2∑

m=1

∑
xy∈E(T2)

(dΩ(x) + dΩ(y))|nx(e|Ω2)(x)− ny(e|Ω2)(y)|,

= |{E(T1) + 1} − {E(T2) + 1}|+
2∑

m=1

∑
xy∈E(T1)

|nx(e|Ω1(x)− ny(e|Ω1)(y)|

− |{E(T1) + E(T1) + 1} − {1}| −
2∑

m=1

∑
xy∈E(T2)

|nx(e|Ω2)(x)− ny(e|Ω2)(y)|,

= −|E(T1)| − |E(T2)|,

There are two cases, if

Case 1. |E(T1)| ≥ |E(T2)|, then −2|E(T2)| < 0,

Case 2. |E(T2)| ≥ |E(T1)|, then −2|E(T1)| < 0,

In each case,
Mowv (Ω1)−Mowv (Ω2) < 0,

which shows maximum weighted vertex moster index. This completes the proof.

Further, we deduce new Ω2 from Ω1 moving all the pendent vertices to a central vertices
such that new graph has maximum weighted vertex moster index.

Lemma 3. Let T1 and T2 be two subgraphs constructed by adjoining n − d − 1 pendent
vertices at central vertices with d (with even diameter). Consider that two subgraphs T1 and
T2 with common ukvk edge between them, presented by Ω1. Now construct Ω2 from Ω1 by
removing all pendent vertices and ukvk edge to central vertex. Then Mowv (Ω1) < Mowv (Ω2).

Proof. Suppose T1 and T2 be subgraphs of Ω1. By construction of Ω2, the num-
ber of closer vertices of end vertices of the fixed edge of T1 and T2 in Ω1 remains same
in Ω2 respectively. Therefore, for an edge xy ∈ E(T ), nx(e|Ω1)(x) = nx(e|Ω2)(x) and
ny(e|Ω1)(y) = ny(e|Ω2)(y) The cut edge ukvk in Ω1 follows as,



F. Asmat et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1794-1808 1800

Case 3. duk
= dvk = (n − d) + 1, and nuk

(e|Ω1)(uk) = [E(T1) + 2, nvk(e|Ω1)(vk) =
E(T2) + 2.

Furthermore, pendent edges at central vertices in Ω1 connected with uk are following
as:

Case 4. duk
= dvk = (n−d)+1, and nuk

(e|Ω1)(uk) = nvk(e|Ω1)(vk) = [E(T )+E(T )]+3,
da∗k = db∗k = 1, for some 1 ≤ k ≤ n− d− 1.

Similarly, 2n − 2d − 1 pendent vertices at central vertex in Ω2 by removing pendent
vertices and cut edge from Ω1. Now, the pendent edges uv∗k, where 1 ≤ k ≤ 2n − 2d − 1
at central vertex of Ω2.

Subcase 1. nu(e|Ω2)(u) = |E(T1) + 2|, and nv∗k
(e|Ω2)(v

∗
k) = 1.

By combining Cases 3,4 and subcase 1, and employing definition of weighted vertex
Mostar index, we have

Mowv (Ω1)−Mowv (Ω2) =

[ 2∑
m=1

∑
xy∈E(T1)

(dΩ1(x) + dΩ1(y))|nx(e|Ω1)(x)− ny(e|Ω1)(y)|

+

n−d−1∑
k=1

(dΩ1(a
∗
k) + dΩ1(uk))|na∗k

(e|Ω1)(a
∗
k)− nuk

(e|Ω1)(uk)|

+
n−d−1∑
k=1

(dΩ1(b
∗
k) + dΩ1(vk))|nb∗k

(e|Ω1)(b
∗
k)− nvk(e|Ω1)(vk)|

]

−
[ 2∑
m=1

∑
xy∈E(T1)

(dΩ2(x) + dΩ2a(y))|nx(e|Ω2)(x)− ny(e|Ω2(y)|

−
2n−2d−1∑

k=1

(dΩ2(u) + dΩ2(v
∗
k))|nv∗k

(e|Ω2)(v
∗
k)− nu(e|Ω2(u)|

]
,

≤
[
{(n− d) + 2}(E(T1)− E(T2)− 2) + {(n− d) + 2}(E(T1) + E(T2) + 1)

]
,

=

[
{(n− d) + 2}(−1)

]
,

< 0

It shows, Mowv (Ω1)−Mowv (Ω2) < 0, which shows proof is complete.

Corollary 2. Consider Ω be the graph with even diameter and T be subgraph of Ω, which
is presented by Ω1. To construct Ω2 from Ω1, remove both pendent vertices connected with
cut edge and attach at central vertex of T . Then Mowv (Ω1) < Mowv (Ω2)



F. Asmat et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1794-1808 1801

Proof. Suppose two subgraphs in Ω1 say, T1 and T2. By construction of Ω2, the
number of closer vertices of end vertices of the fixed edge of T1 and T2 in Ω1 remains
same in Ω2, respectively. Therefore, for an edge xy ∈ E(Tr), where r ∈ {1, 2} we have
nx(e|Ω1)(x) = nx(e|Ω2)(x) and ny(e|Ω1)(y) = ny(e|Ω2)(y). The number of closed vertices
of a fixed vertex of T in Ω1 is d and d−1 in Ω2. The following case for cut edge uv ∈ E(Ω1).

Case 5. du = dv = 3, and nu(e|Ω1)(u) = E(T ) + 1, nv(e|Ω1)(v) = 3.

Similar results for pendent vertices adjoining at vertex u.

Case 6. nv(e|Ω1)(v) = E(T ) + 3, and nu∗
1
(e|Ω1)(u

∗
1) = nu∗

2
(e|Ω1)(u

∗
2) = 1.

There are following cases in Ω2, for pendent edges uv ∈ E(Ω2).

Case 7. du = 3, dv = du∗
2
= du∗

1
= 1.

Case 8. nu(e|Ω2)(u) = E(T ) + 3, and nv(e|Ω2)(v) = nu∗
1
(e|Ω2)(u

∗
1) = nu∗

2
(e|Ω2)(u

∗
2) = 1.

By combining Cases 5, 6, 7 and 8, and employing definition of weighted vertex Mostar
index, we have,

Mowv (Ω1)−Mowv (Ω2) =

[ 2∑
m=1

∑
xy∈E(T1)

(dΩ1(x) + dΩ1(y))|nx(e|Ω1)(x)− ny(e|Ω1)(y)|

+ (dΩ1(u) + dΩ1(v))|nv(e|Ω1)(v)− nu(e|Ω1)(u)|
+ (dΩ1(u

∗
1) + dΩ1(v))|nu∗

1
(e|Ω1)(u

∗
1)− nv(e|Ω1)(v)|

+ (dΩ1(u
∗
2) + dΩ1(v))|nu∗

2
(e|Ω1)(u

∗
2)− nv(e|Ω1)(v)|

− (dΩ2(v) + dΩ2(u))|nv(e|Ω2)(v)− nu(e|Ω2)(u)|
− (dΩ2(u

∗
1) + dΩ2(u))|nu∗

1
(e|Ω2)(u

∗
1)− nu(e|Ω2)(u)|

− (dΩ2(u
∗
2) + dΩ2(u))|nu∗

2
(e|Ω2)(u

∗
2)− nu(e|Ω2)(u)|

−
2∑

r=1

∑
xy∈E(T2)

(dΩ2(x) + dΩ2(y))|nx(e|Ω2)(x)− ny(e|Ω2)(y)|
]
,

=

[
(3 + 3)(|3− |E(T )| − 1|) + (3 + 1)(|1− |E(T )| − 3|) + (3 + 1)(|1− |E(T )| − 3|)

− (3 + 1)(|1− |E(T )| − 3|)− (3 + 1)(|1− |E(T )| − 3|)− (3 + 1)(|1− |E(T )| − 3|)
]
,

≤ −6|E(T )|+ 4|E(T )|+ 20,

< 0

It shows, Mowv (Ω1) −Mowv (Ω2) < 0, which shows the maximum weighted vertex Mostar
index.



F. Asmat et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1794-1808 1802

Lemma 4. Consider Ω be the graph with even diameter and T be subgraph of Ω, which
is presented by Ω1. To construct Ω2 from Ω1, remove all n − d − 1 pendent vertices and
identifying at central vertex of T . Then Mowv (Ω1) < Mowv (Ω2).

Proof. By use of corollary 2, we can easily extend the graph up-to n− d− 1 pendent
vertices linked with cut edge and proof is obvious.

Next, we turn to the proof of Theorem 3.
Proof. Assume Ω ∈ T (n, d) be a graph with d ≥ 1 and n ≥ 2. If T (n, d) ≇ Ω and Ω

has cut edge then repeatedly by using Lemma 2, we acquire sequence of new tree graphs
Ω1,Ω2,Ω3, ...,Ωα, where Ωα be a tree graph without edge with largest degree sequence such
that Mowv (Ω1) < Mowv (Ω2) < Mowv (Ω3) < ... < Mowv (Ωα). Now, If Mowv (Ωα) ≇ T (n̂, d)
and Mowv (Ωα) has subgraph with n − d − 1, pendent vertices with even diameter then
repeatedly using Lemma 3, we can acquire sequence of tree graph such that Ωα1 ,Ωα2 ,Ωα3 ,..,
Ωαβ

, satisfying Mowv (Ωα1) < Mowv (Ωα2) < Mowv (Ωα3) < ... < Mowv (Ωαβ
), where Ωαβ

be a
tree graph such that degree of central vertex with even diameter greater than 3.

If Ωαβ
≇ T (n̂, d), then repeatedly using Lemma 4, and corollary 2, we have Ωαβ1

,Ωαβ2
,Ωαβ3

, ...,Ωαβγ
,

satisfyingMowv (Ωαβ1
) < Mowv (Ωαβ2

) < Mowv (Ωαβ3
) < ... < Mowv (Ωαβγ

), whereMowv (Ωαβγ
) ∼=

T (n̂, d).
This completes the proof.

Transformation 1. Suppose that k1 and k2 are two graphs with hl ∈ VΩ(kl) for l = {1, 2}.
Suppose that Pn, Tn, Sn are path, tree and star of the same order n such that Tn ≇ Pn

and Tn ≇ Sn. Let Ω be a generated graph by associating the node h1 with one end of Pn

and associating the node h2 with the other end of Pn.
Next, Let Ω

′
be a transformed graph by associating the vertex h1 and h2 as a new vertex

h, and then fix the vertex h with the center of Sn.

Lemma 5. We say that Ω and Ω
′
are graphs.Then Mowe (Ω) < Mowe (Ω

′
).

Proof. Since Tn ≇ Pn and Tn ≇ Sn, n ≥ 4
For any edge e = gh1 ∈ E(k1), mg(e|Ω) − mh1(e|Ω) = mg(e|Ω

′
) − mh1(e|Ω

′
) and

dΩ(g) + dΩ(h1) ≤ dΩ′ (g) + dΩ′ (h1). Therefore,∑
e=gh1∈E(k1)

dΩ(g) + dΩ(h1)|mg(e|Ω)−mh1(e|Ω)|,

<
∑

e=gh1∈E(k1)

dΩ′ (g) + dΩ′ (h1)|mg(e|Ω
′
)−mh1(e|Ω

′
)|,

Similarly, for any edge e = g
′
h2 ∈ E(k1), mg′ (e|Ω)−mh2(e|Ω) = mg′ (e|Ω

′
)−mh2(e|Ω

′
)

and dΩ(g
′
) + dΩ(h2) ≤ dΩ′ (g

′
) + dΩ′ (h2). Therefore,∑

e=g′h2∈E(k1)

dΩ(g
′
) + deΩ(h2)|mg′ (e|Ω)−mh2(e|Ω)|,



F. Asmat et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1794-1808 1803

<
∑

e=g′h2∈E(k1)

dΩ′ (g
′
) + dΩ′ (h2)|mg′ (e|Ω

′
)−mh2(e|Ω

′
)|,

For any edge e = hk ∈ E(Pn), mh(e|Ω) − mk(e|Ω) = |Ω|. Similarly, e = hk ∈ E(Sn),
mh(e|Ω

′
)−mk(e|Ω

′
) = |Ω′ |. Therefore,∑
e=hk∈E(Pn)

dΩ(h) + dΩ(k)|mh(e|Ω)−mk(e|Ω)|,

<
∑

e=hk∈E(Sn)

dΩ′ (h) + dΩ′ (k)|mh(e|Ω
′
)−mk(e|Ω

′
)|,

Which is obvious to show that Mowe (Ω) < Mowe (Ω
′
) < 0. The proof is complete.

Transformation 2. Let K be a graph with u
′ ∈ V (K), and Cp be a cycle of order p. The

graph K(u
′
)Cp is constructed by associating the vertex u

′
with a vertex of Cp. Let Ω be

a graph generated from K(u
′
)Cp by connecting pendent edges to the vertices of Cp other

than u
′
.

Next, Let Ω
′
be a generated graph from Ω by moving all pendent edges, which are

attached at vertices of Cp other than u
′
, on u

′
. Given that, |Ω| = |Ω′ |.

Lemma 6. Suppose that two graphs denoted by Ω and Ω
′
. Then Mowe (Ω) < Mowe (Ω

′
).

Proof. Suppose |Ω| = |Ω′ | = n. In Ω and Ω
′
, suppose the vertices of Cp are

u
′
o(u

′
)u

′
1u

′
2, .., u

′
p−1 subsequently. In Ω, suppose that yj pendant edges rooted on u

′
j for

1 ≤ j ≤ p− 1 and

p−1∑
j=1

yj = y.

Since, dΩ(u
′
) = dΩ(u

′
) − y. For any edge e = wu

′ ∈ E(K), dΩ(w) + dΩ(u
′
) <

dΩ′ (w) + dΩ′ (u
′
), and nw(e|Ω)− nu′ (e|Ω) = nw(e|Ω

′
)− nu′ (e|Ω′

). Therefore,

=
∑

wu′∈E(K)

(dΩ(w) + dΩ(u
′
))|nw(e|Ω)− nu′ (e|Ω)|,

−
∑

wu′∈E(K)

dΩ′ (w) + deΩ′ (u
′
)nw(e|Ω

′
)− nu′ (e|Ω′

),

< 0

In Ω, for pendant edge, e = uju
′
j rooted on u

′
j (1 ≤ j ≤ p − 1), dΩ(uj) + dΩ(u

′
j) =

yj + 2 + 1 = yj + 3 and nuj (e|Ω)− n
u
′
j
(e|Ω) = n− d.

p−1∑
j−1

∑
uju

′
j∈E(Ω)

dΩ(uj) + dΩ(u
′
j)|nuj (e|Ω)− n

u
′
j
(e|Ω)| = (n− d)

p=1∑
j=1

yj(yj + 3),



F. Asmat et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1794-1808 1804

In Ω
′
, for pendant edge e = uu

′
rooted on u

′
which is not in E(K) and dΩ′ (u) = 1,

dΩ′ (u) + dΩ′ (u
′
) = dΩ(u

′
) + y + 1 and nu(e|Ω

′
)− nu′ (e|Ω′

) = n− d.∑
uu′∈E(Ω′ )

dΩ′ (u) + dΩ′ (u
′
)|nu(e|Ω

′
)− nu′ (e|Ω′

)| = y(n− d)(dΩ + y + 1),

There is two possibilities, either p is even or p is odd. First we suppose p is even. For
any edge e = u

′
ju

′
j+1 (0 ≤ j ≤ p − 1) of Cp in Ω, dΩ(u

′
j) + dΩ(u

′
j+1) = yj + yj+1 + 4

when 1 ≤ j ≤ p− 2, dΩ(u
′
0) + dΩ(u

′
1) = dΩ(u

′
) + y1 + 2 when j = 0, dΩ(u

′
p−1) + dΩ(u

′
0) =

dΩ(u
′
) + yp−1 + 2 when j = p− 1. Since p is even n

u
′
j
(e|Ω)− n

u
′
j+1

(e|Ω) = n− d.

=

p−1∑
j=0

dΩ(u
′
j) + dΩ(u

′
j+1)|nu

′
j
(e|Ω)− n

u
′
j+1

(e|Ω)|,

= 2(n− d)(dΩ(u
′
) + y + 2(p− 1)),

Similarly in Ω
′
, after simplification we have,

=

p−1∑
j=0

dΩ′ (u
′
j) + dΩ′ (u

′
j+1)|nu

′
j
(e|Ω′

)− n
u
′
j+1

(e|Ω′
)|,

= 2(n− d)(dΩ′ (u
′
) + 2(p− 1))

= 2(n− d)(dΩ′ (u
′
) + y + 2(p− 1)),

By using the above calculations, we have straightforward result

Mowe (Ω)−Mowe (Ω
′
) =

p−1∑
j=0

dΩ(u
′
j) + dΩ(u

′
j+1)|nu

′
j
(e|Ω)− n

u
′
j+1

(e|Ω)|,

−
p−1∑
j=0

dΩ′ (u
′
j) + dΩ′ (u

′
j+1)|nu

′
j
(e|Ω′

)− n
u
′
j+1

(e|Ω′
)|,

< 0,

The result Mowe (Ω) < Mowe (Ω
′
) is obvious similar for p is odd. The proof is complete.

Transformation 3. Let K be a graph with u ∈ V (K) such that dK(u) ≥ 2, and Cp be a
cycle of order p such that p ≥ 4. Let Ω be a generated graph from K(u)Cp by combining
a vertices of Cp with u.

Next, Let Ω
′
be the graph obtained from Ω by exchanging Cp for C3 and p− 3 pendent

edges. Note that, |Ω| = |Ω′ |.

Lemma 7. Let Ω and Ω
′
be two connected graphs explained in Transformation 3. Then

Mowe (Ω) < Mowe (Ω
′
).



F. Asmat et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1794-1808 1805

Proof. Suppose |Ω| = |Ω′ | = n without loss of generality. Note that dΩ(u) = dΩ′ (u)−
(p− 3). For any edge e = uu

′ ∈ E(K), dΩ(u) + dΩ(u
′
) < dΩ′ (u) + dΩ′ (u

′
), and mu(e|Ω)−

mu′ (e|Ω) = mu(e|Ω
′
)−mu′ (e|Ω′

). Therefore,

=
∑

e=uu′∈E(K)

dΩ(u) + dΩ(u
′
)|mu(e|Ω)−mu′ (e|Ω)|,

−
∑

e=uu′∈E(K)

dΩ′ (u) + dΩ′ (u
′
)|mu(e|Ω

′
)−mu′ (e|Ω′

)|,

< 0

When p is odd, for the edges in Cp of Ω

=

p−1∑
j=0

dΩ(uj) + dΩ(uj+1)|muj (e|Ω)−muj+1(e|Ω),

= 4(p− 1) + (n− 1)[2(dΩ(u) + 2) + 4(p− 3)],

When p is even, for the edges in Cp of Ω

=

p−1∑
j=0

dΩ(uj) + dΩ(uj+1)|muj (e|Ω)−muj+1(e|Ω)|,

= n[2(dΩ(u) + 2) + 4(p− 2)],

For any pendant edge uu
′
rooted on u in Ω

′
, dΩ′ (u) + dΩ′ (u

′
) = dΩ′ (u) + p − 2, and

mu(e|Ω
′
)−mu′ (e|Ω′

) = n− d and dΩ′ = 1. Therefore,

=
∑

e=uu′∈E(Ω′ )\E(K)

dΩ′ (u) + dΩ′ (u
′
)|nu(e|Ω

′
)− nu′ (e|Ω′

)|,

= (p− 3)n[(dΩ(u)) + (p− 2)],

For the edges in C3 of Ω
′

=

2∑
j=0

dΩ′ (uj) + deΩ′ (uj+1)|muj (e|Ω
′
)−muj+1(e|Ω

′
)|,

= 8 + [2(dΩ′ (u) + 2)(n− 1)],

= 8 + 2(dΩ′ (u) + p− 1)(n− 1),

By using the above calculations, we have straightforward result Mowe (Ω)−Mowe (Ω
′
) < 0,

which completes proof.



REFERENCES 1806

Transformation 4. Assume that K is a graph and Cp is a cycle of order p. The graph
formed by associating the vertex u with a vertex of Cp is denoted by K(u)Cp. Let Ω be
the graph formed from K(u)Cp by attaching some triangles and (or) some pendent edges
to the vertices of Cp except u.

Now, Let Ω
′
be the graph formed from Ω by shifting all triangles and pendent edges

rooted on vertices of Cp except for u to u. Given that |Ω| = |Ω′ |.

Lemma 8. Suppose that Ω and Ω
′
are two graphs. Then Mowe (Ω) < Mowe (Ω

′
).

We leave to the reader the proof of Lemma 8, since it is similar to the proof of Lemma
6.

Transformation 5. Let Ck be a cycle with r ≥ 4 and u
′
, u

′
1, ..., u

′
k−1 are the vertices of

Ck subsequently. Let Ko be a cactus graph such that d(Ko) ≥ 2 and all cycles in Ko are
triangles. Suppose w, x are two vertices of V (Ko) such that w, x are in some triangles of
Ko and d(Ko)(w) = d(Ko)(x) = 2. Let Ω be the resulting graph by associating u

′
and x

via a path (the length of the path ≥ 0), and fix u
′
j where i ̸= 0 with one vertex of a graph

K.
Next, Let Ω

′
be a generated graph from Ω by exclude the edge u

′
jy for any y ∈ VK and

addition of edge wy.

Lemma 9. Suppose Ω and Ω
′
are two graphs. Then Mowe (Ω) > Mowe (Ω

′
).

The proof of Lemma 9, leave to readers, since it is evidently analogous to the proof of
Lemma 7.

Next, we turn to the proof of Theorem 4.
Proof. By using Lemmas 1, 5, 6, 7, 8, and 9 for any graph Ω ∈ C(n, k), Mowe (Ω) <

Mowe (S
∗,k
n ) and the equality holds if and only if Ω ∼= S∗,kn . It is easy to compute that

Mowe (Ω) = n3 + (k − 4)n2 + (−3k + 5)n− (2n+ 4). The proof completes.

Acknowledgements

The authors are grateful to two anonymous referees for their valuable comments and correc-
tions, which have led to considerable improvement of the presentation of this work. This project
is funded by King Saud University, Riyadh, Saudi Arabia. Research Supporting Project number
(RSP2023R167), King Saud University, Riyadh, Saudi Arabia.

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