Abstract: In this work, we introduce a topology Ttot

G, called the total graphic topology, for the vertices

set of a directed graph G= (V, E). We prove many properties of this topology and we give some open

sets and some closed ones. We prove that Ttot

Gis an Alexandro topology. In addition, we investigate

functions between directed graphs, the connectedness of this topology for some strongly connected graphs

and we give an example for each case.

Key-Words: Directed graph, topology, minimal bases, connected components, homeomorphism.

Received: April 14, 2023. Revised: February 19, 2023. Accepted: March 14, 2024. Published: April 22, 2024.

1 Introduction

Graph theory attires attention sice the resolving

of the problem of the Königsberg seven bridges,

[1]. It becomes one of discrete mathematics struc-

tures.

Graphs are simple to understand and can be used

for representing many of mathematical combina-

tions. Today, graph theory becomes a fundamen-

tal mathematical tool for many domains as chem-

istry, marketing and computers network. If we add

topology to the graph, we can use them to solve

economic and the track ow problems, [2], [3],

[4], as in medical application and blood circula-

tion, [5], [6], [7], [8].

A topology is called an Alexandro topology if any

intersection of open sets is also an open set, [9],

[10]. Such topology is very interesting because we

have a minimal bases and the characteristic prop-

erties can be studied by using this minimal bases

or its subbases.

In, [11], the graphic topology for undirected graph

was introduced on the vertices set. After that

many topologies are introduced on undirected

graphs.

In, [12], the authors investigated the graphic

topology and solved partially an open problem

mentioned in, [11]. After that, in 2023 the Z-

graphic topology was introduced in order to an-

swer the rst open problem in, [11], and bypass

it, see, [13]. Graphic topology was also dened on

fuzzy graphs in, [14].

For directed graphs in, [15], two topologies on the

edges set are given. In this paper, we consider

directed graph and introduce the total graphic

topology on the vertices set.

This work has ve sections in addition to the in-

troduction and conclusion. Section 2 is devoted to

some useful preliminaries in directed graph theory

and topology. We give a set of subset of the ver-

tices set Vof a directed graph G= (V, E)which

is will be the subbases of our graphic topology. In

section 3, we prove a lot of typical and preliminary

results as proving that the total graphic topol-

ogy is an Alexandro topology, we prove some

characterizations of minimal open sets. Section

4 is devoted to some advantaged results and we

give some examples of open and closed sets. In

Section 5, we study functions between digraphs

and their relation with continuous and homeomor-

phism maps. The last section is devoted to total

graphic topology and connectedness.

2 Preliminaries

In this section, we will recall some denitions and

properties of directed graph theory and topology,

[16], [17], [18]. Then, we introduce a new topology

for the graph which will be have the total graphic

topology as name.

Denition 2.1 A directed graph G= (V, E)is a

pair of sets: a nonempty set Vand a set Esuch

Total Graphic Topology on the Vertex Sets of Directed Graphs

HANAN OMAR ZOMAM

Department of Mathematics

College of Science Al-Madinah Al-Munawarah

Taibah University

SAUDI ARABIA.

Department of Mathematics

Faculty of Science

Shendi University

SUDAN.

PROOF

DOI: 10.37394/232020.2024.4.3

Hanan Omar Zomam

E-ISSN: 2732-9941

Volume 4, 2024

that E⊂V×V. More precisely, if e= (x, y)∈E,

ehas a direction from xto y. We also say eis an

edge from xto y.

A directed graph G= (V, E)is called simple if

∀x∈V, (x, x) /∈Eand ∀x, y ∈Vthere is no

multiple edges from xto y.

Denition 2.2 A digraph G= (V, E)is called

complete if it is simple and for any distinct x, y ∈

V, there exist a unique edge from xto yand a

unique edge from yto x.

Denition 2.3 Let G= (V, E)be a digraph. G

is called an oriented graph if at most one of the

two edges (x, y)and (y, x)is in E, for all xand y

vertices of G.

If Gis a simple graph and (x, y)∈Eif and only

if (y, x) /∈E, for all x, y ∈V, we call Gis a

tournament.

Denition 2.4 Let G= (V, E)be a simple di-

graph. The digraph G= (V, E)dened by

(x, y)∈Eif and only if (x, y) /∈Eis called the

complement of G.

Denition 2.5 Consider a directed graph G=

(V, E).

In G, a directed path Pfrom a0to anis a sequence

of the form P:a0, e0, a1, e1,· · · , an−1, en−1, an,

where ak∈Vand ekan edge from akto ak+1,

k= 0,· · · , n −1.

We say that aand bare connected in Gif there

is a directed path from ato band a directed path

from bto a. Also, Gis called strongly connected

if any two distinct vertices are connected in G.

Let xa vertex of a simple directed graph G=

(V, E). We dene the out-neighborhood set of x

KHx={y∈V, (x, y)∈E}.(1)

and the int-neighborhood set of xas

Dx={y∈V, (y, x)∈E}.(2)

It is clear from (1) and (2) that

y∈ KHxif and only if x∈ Dy.

Let

Mx=KHx∪ Dx.(3)

The cardinal of the out-neighborhood KHxof xis

called the out-degree of x, we denote

d+(x) = card(KHx),(4)

the cardinal of int-neighborhood Dxof xis named

int-degree of the vertex xand we set

d−(x) = card(Dx)(5)

Figure 1: A directed graph G= (V, E)

and we donate

dt(x) = card(Mx)(6)

the total degree of the vertex x.

We set the minimum out-degree, the minimum int-

degree and the minimum total degree of a digraph

G= (V, E)as

δ+(G) = min{d+(x), x ∈V},(7)

δ−(G) = min{d−(x), x ∈V}(8)

and

δt(G) = min{dt(x), x ∈V}(9)

However, the maximum out-degree, the maximum

int-degree and the maximum total degree of Gare

respectively given by

∆+(G) = max{d+(x), x ∈V},(10)

∆−(G) = max{d−(x), x ∈V}(11)

and

∆t(G) = max{dt(x), x ∈V}.(12)

Example 2.1 For the graph Ggiven by Fig. 1, we

have Ma={b, c, d},Mb={a, c},Mc={a, b}

and Md={a}.

Denition 2.6 Suppose that G= (V, E)is a di-

graph. A vertex x∈Vis called isolated if

Mx=∅.

Denition 2.7 Let G= (V, E)be a digraph. We

say that Gis locally nite if Mxis a nite set for

all x∈V.

It is clear that a nite digraph is locally nite one.

We pass to give some denitions and notations for

topological spaces that we will need them later.

Denition 2.8 Let Vbe a non empty set and let

τbe a family of subsets of V. If the following

conditions

PROOF

DOI: 10.37394/232020.2024.4.3

Hanan Omar Zomam

E-ISSN: 2732-9941

Volume 4, 2024

(1) ∅, V ∈τ;

(2) ∀U1, U2∈τ, we have U1∩U2∈τ;

(3) ∀ {Ui}i∈Ia family of elements in τ, the union

∪i∈IUi∈τ

are satised, we say τis a topology for Vor (V, τ )

is a topological space.

An element of τis named an open set of V.

Denition 2.9 Suppose that (V, τ )is a topological

space and U⊂V.

(i) The set Uc=V\Uis called the complement

of Uin V.

(ii) The set Uis called a closed set of Vif and

only if Ucis an open set.

(iii) We denote Uthe smallest closed set of Vcon-

taining U.Uis called the closure of Uin V.

Next, suppose that the digraph G= (V, E)is sim-

ple and without isolated vertices. Consider the set

Stot

G={Mx;x∈V},(13)

where Mxis given by (3).

Theorem 2.1 Let G= (V, E)be a simple directed

graph without isolated point. Then, Stot

Gis a sub-

bases for a topology of the vertices set V.

Proof. Since Sx∈VMx⊂V, we have to prove

that V⊂Sx∈VMx.

Let z∈V, Since Mz6=∅, there exists x∈ Mz.

So, z∈ Mxand we get the result.

The topology induced by the subbases Stot

Gis

called the total graphic topology of Gand it is

denoted Ttot

Example 2.2 For the graph in the Example 2.1,

we have Stot

G=n∅,{b, c, d},{a, c},{a, b},{a}o,

B=n{a},{b},{c},{a, b},{a, c},{b, c, d}oand

Ttot

G=n∅,{a},{b},{c},{a, b},{a, c},{b, c},

{a, b, c},{b, c, d},{a, b, c, d}o.

Throughout this paper, a digraph means a simple

locally nite digraph without isolated vertex.

3 Preliminary Results

An Alexandro space is a topological space sat-

isfying any intersection of open sets is an open

set. With an Alexandro space and so Alexandro

topology we have a minimal bases of the topology

and we can use it to study the properties of the

topological space as we will do in the rest of this

paper.

Theorem 3.1 suppose that G= (V, E)is a di-

rected graph. Then (V, Ttot

G)is an Alexandro

space.

Proof. The total graphic topology Ttot

Gis con-

structed from the subbases Stot

G, so it is sucient

to prove that any intersection of elements in the

subbases is an open set.

Consider ∩x∈AMx, where A⊂V, and suppose

that \

x∈A

Mx6=∅.

Suppose that y∈ ∩x∈AMx. Then, y∈ Mx, for

all x∈A.

We get for all x∈A,x∈ My. Therefore

this means A⊂ Mxand so Ais nite. Hence

∩x∈AMxis an open set.

As consequence of the above theorem, Let G

be a digraph, then the total graphic topology Ttot

of a directed graph G= (V, E)has a minimal

basis which we denote

BG={Mx;x∈V},(14)

where Mxis the intersection of all open sets con-

taining x, it is the smallest open set containing the

vertex x. We can characterise the smallest open

sets by using the subbases as follows.

Theorem 3.2 Suppose that G= (V, E)is a di-

graph and xis a vertex of G. Then,

Mx=\

z∈Mx

Mz.(15)

Proof. Since xis a vertex of G,Mxis a nonempty

set. Consider z∈ Mx, then x∈ Mzand so the

open set ∩z∈MxMzcontains xand so

Mx⊂\

z∈Mx

Mz.

Conversely, since Ttot

Ghas Stot

Gas subbases there

exists A⊂Vsuch that Mx=∩z∈AMz.

For all z∈A,x∈ Mz. Therefore, for all z∈A,

z∈ Mx. Then, A⊂ Mxand so,

z∈Mx

Mz⊂\

z∈A

Mz=Mx.

Remark 3.1 For a directed G= (V, E), each min-

imal open set Mxis a nite set since each set Mz

is nite et

Mx=\

z∈Mx

Mz.

PROOF

DOI: 10.37394/232020.2024.4.3

Hanan Omar Zomam

E-ISSN: 2732-9941

Volume 4, 2024

Corollary 3.1 Let Gbe a digraph and let xand a

two distinct vertices of G.

(i) If Mx={a}, then Mx=Ma.

(ii) If a∈ Mx, then Mx⊂ Ma.

(iii) If Ma⊂ Mx, then Mx⊂ Ma.

Proof.

(i) If Mx={a}, from the fact that Mx=

∩z∈MxMzwe get Mx== Ma.

(ii) From the last theorem, Mx=∩z∈MxMz, so

Mx⊂ Mzfor all z∈ Mx. In particular,

Mx⊂ Ma.

(iii) When Ma⊂ Mx, we have a∈Ma⊂ Mx.

The result follows From (ii).

The following result follows from the Theorem 3.2,

but it very useful when dealing with the minimal

bases.

Theorem 3.3 When G= (V, E)is a digraph and

aa vertex of G. Then,

Ma={x∈V;Ma⊂ Mx}.

Proof. We have

Ma=\

z∈Ma

Mz,

so we have x∈Maif and only if x∈ Mz,∀z∈ Ma

which is equivalent to for all z∈ Ma,z∈ Mxand

this is true if and only if Ma⊂ Mx.

Corollary 3.2 Suppose that Gis a digraph and let

abe a vertex of G. Then, Ma∩ Ma=∅. Also, if

Mx⊂ Ma, we have Mx∩Ma=∅.

Proof. (i)By contradiction, suppose that there

exists y∈Ma∩ Ma.

y∈Magives Ma⊂ Myfrom Theorem 3.3.

y∈ Maimplies y∈ My, contradiction since Gis

a simple directed graph.

(ii)If Mx⊂ Ma, then Mx∩Ma⊂ Mx∩Ma=∅.

Theorem 3.4 For any vertex ain a directed graph

G= (V, E), we have

{a}={x∈V;Mx⊂ Ma}.

Proof. x∈ {a}if and only if A∩ {x} 6=∅, for

all open set Acontaining x. Using minimal bases,

this is equivalent to Mx∩ {a} 6=∅, this means

a∈Mx. That is, by Theorem3.3

x∈ {a} ⇔ Mx⊂ Ma.

4 Some Properties of Graphic

Topology

Theorem 4.1 For a directed graph G= (V, E, the

space (V, Ttot

G)is a compact topological space if

and only if the vertices set Vis nite.

Proof. Recall that the topological space (V, Ttot

is said compact if every open cover of the space

V⊂Sx∈VUxhas a nite subcover.

If Vis a nite set, then from any open cover of

V, we have a nite subcover by denition of the

compactness.

Conversely, suppose that (V, Ttot

G)is a compact

topological space. Consider the minimal basis BG.

The family BGis an open cover of V, so there

exists a nite subcover of BG. Since it is minimal

as basis, BGis equal to this subcover. Since from

(14), we have

BG={Mx;x∈V},

we conclude that Vis nite.

Proposition 4.1 Let G= (V, E)be a digraph.

Then, A={x∈V, dt(x)=∆t(G)}is an open

set for the total graphic topology of G.

Proof. Let x∈A. We will prove that Mx⊂A.

Let y∈Mx, we have Mx⊂ My(from Theorem

3.3). We obtain

card(Mx)≤card(My)≤∆t(G).

Then dt(x) = ∆t(G) = dt(y)and hence y∈A.

We get x∈Mx⊂A, for all x∈Aand the result

is proved.

Proposition 4.2 Let G= (V, E)be a digraph.

Then the following set

B={x∈V, dt(x) = δt(G)}(16)

is a closed set for the total graphic topology of G.

Proof. We have B⊂B. We will prove the inverse

inclusion.

Let x∈B, since Mxis an open set containing x,

we have Mx∩B6=∅.

Set z∈Mx∩B, we obtain the two facts:

Mx⊂ Mzand dt(z) = δt(G).

So,

card(Mx)≤card(Mz) = δt(G).

Therefore dt(x) = δt(G)and we get x∈B.

Proposition 4.3 Suppose that G= (V, E)is a -

nite directed graph. Then the following set

G={A;Ac∈ T tot

G}(17)

is a topology for Vand if Gis an oriented graph

such that Ttot

G=Tc

G, then Ttot

Gis the discrete

topology.

PROOF

DOI: 10.37394/232020.2024.4.3

Hanan Omar Zomam

E-ISSN: 2732-9941

Volume 4, 2024

Proof. We have ∅, V ∈τsince their complements

are in Ttot

When Aand Bare in Tc

G, we have (A∩B)c=

Ac∪Bcand so (A∩B)c∈ T tot

G. Hence A∩B∈ T c

Now, if we have a countable family {Ai}of ele-

ments of Tc

G, we know that

(∪iAi)c=∩iAc

But Ttot

Gis an Alexandro topology, we deduce

that ∪iAi∈ T c

Then Tc

Gis a topology for V.

Next, suppose that Gis an oriented graph

and Ttot

G=Tc

G. Let x∈V, we have Mx∪ {x}c

as the set of all vertices adjacent to xin G. Then,

Mx∪ {x}c∈ T tot

Gsince it is an element of its

subbases.

Therefore, Mx∪ {x}c∈ T c

Gand so,

Mx∪ {x} ∈ T tot

Since Mxis the minimal open set containing x, we

get

Mx⊂ Mx∪ {x}.

Since Mx∩ Mx=∅(Corollary 3.2), we obtain

Mx={x}. We have so Ttot

Gis the discrete topol-

ogy.

5 Isomorphic Digraphs and

Homeomorphic Graphic Topologies

Denition 5.1 Un homomorphism hfrom a di-

graph G= (V, E)to a digraph G0= (V0, E0)is

a function h:V→V0satisfying

∀(x, y)∈E, (h(x), h(y)) ∈E0.

his called isomorphism if h:V→V0is bijective

and ∀(x, y)∈V2, we have

(x, y)∈Eif and only if (h(x), h(y)) ∈E0.

We say that the two graphs are isomorphic.

Denition 5.2 An homeomorphism hfrom a topo-

logical space (V, T )to a topological space (V0, T 0)

is a continuous bijective map h:V→V0such

that its inverse is also continuous. In this case,

the two spaces or the two topologies are called

homeomorphic.

Our rst result in this section is the following.

Theorem 5.1 Suppose that two directed graphs

G= (V, E)and G0= (V0, E0)are isomorphic and

h:V→V0is an isomorphism. Then, their total

graphic topologies are homeomorphic.

Figure 2: These two graphs have the discrete

topology as total graphic topology but they are

not isomorphic.

Proof. It is sucient to prove that for all A∈ Stot

G0,

h−1(A)is in Ttot

G. For this, let z∈V0satisfying

A=Mzand let x=h−1(z).

In this case, we get

h−1(A) = {a∈V;h(a)∈ Mz}

={a∈V; (z, h(a)) ∈E0or (h(a), z)∈E0}

={a∈V; (h(x), h(a)) ∈E0or (h(a), h(x)) ∈E0}

={a∈V; (x, a)∈Eor (a, x)∈E}

=Mx∈ T tot

Hence the bijective function h:V→V0is

continuous. In a similar way, we prove that h−1

is continuous.

For the converse, see Fig. 2. The two graphs

have the the discrete topology as total graphic

topology but they are not isomorphic.

Theorem 5.2 Suppose that G= (V, E)and G=

(V0, E0)are two digraphs and h:V→V0is a

function. Then, his continuous if and only if

∀y, z ∈V, My⊂ Mz=⇒ Mh(y)⊂ Mh(z).

(18)

Proof. If the function his a continuous function

and y, z ∈Vsuch that

My⊂ Mz.

In order to get Mh(y)⊂ Mh(z), we are going to

prove that h(z)∈Mh(y)and the result follows

from Theorem 3.3.

Now, consider the minimal open set Mh(y). such

that y∈h−1Mh(y)and Mythe smallest open

set containing y, we get My⊂h−1Vh(y).

As My⊂ Mz, we obtain z∈Myand so

z∈h−1Mh(y), this means, h(z)∈Mh(y).

For the converse, Suppose that the property

(18) is satised and we have to prove that the

function his continuous. Let Aan open set of V0

PROOF

DOI: 10.37394/232020.2024.4.3

Hanan Omar Zomam

E-ISSN: 2732-9941

Volume 4, 2024

and consider y∈h−1(A). We have h(y)∈Aand

so Mh(y)⊂A.

Let z∈My, then My⊂ Mzfrom The-

orem 3.3. Therefore, Mh(y)⊂ Mh(z)and

hence h(z)∈Mh(y). Or, we have Mh(y)⊂A

then h(z)∈Aand so z∈h−1(A). We get

My⊂h−1(A), for all y∈h−1(A)and the result

follows.

Theorem 5.3 Suppose that G= (V, E)and G0=

(V0, E0)are two digraphs and h:V→V0a func-

tion. Then, h: (V, Ttot

G)→(V0,Ttot

G0)is an home-

omorphism if and only if

∀y, z ∈V, My⊂ Mz⇐⇒ Mh(y)⊂ Mh(z).

(19)

Proof. First, by the Theorem 5.2, we have: h

continuous if and only if

∀y, z ∈V, My⊂ Mz=⇒ Mh(y)⊂ Mh(z).

Using Theorem 5.2 for the function h−1, we get:

h−1continuous if and only if

∀y0, z0∈V0,My0⊂ Mz0=⇒ Mh−1(y0)⊂ Mh−1(z0).

Since his bijective, we get: h−1continuous if and

only if

∀y, z ∈V, Mh(y)⊂ Mh(z)=⇒ My⊂ Mz

and so, the result follows.

6 Graphic Topology and

Connectedness

Being connected, is a property can be dened for a

topological space as for a graph. Here, we will con-

sider the strongly connectivity for directed graph,

[16], [17], [18]. Let us recall the denitions.

Denition 6.1 Let (V, T)be a topological space.

The space Vis said connected if whenever

V=A∪Bsuch that A∩B=∅, we have necessary

A=∅or B=∅. That is, Vcan not written as

the union of two disjoint proper open sets.

Denition 6.2 Let G= (V, E)be a digraph. Gis

called strongly connected if for all a, b ∈Vthere

exist at least two paths joining aand b: one from

ato band one from bto a.

In general, a digraph G= (V, E)does not have

to be strongly connected, so we can dene their

connected components.

Denition 6.3 Suppose that G= (V, E)is a di-

graph. Let U1, U2,· · · be subsets of Vsuch that

Figure 3: This is a non strongly connected digraph

but its total graphic topology is connected.

(i) V=∪iUi;

(ii) Ui∩Uj=∅, for all i6=j;

(iii) For i= 1,2,· · · , for all a, b ∈Ui, there exist a

path from ato band a path from bto a.

(iv) For all a∈Ui,b∈Ujand i6=j, if there exists

a path from ato b, then there is no path from

bto a.

Then, each subset Uiis called connected compo-

nent of the digraph G.

It is clear that a strongly connected digraph has

one connected component. and a nite digraph

has a nite connected components. When the

graph is undirected and disconnected, the graphic

topological is disconnected and this is due to the

connected components are open sets [11], but for

directed graph this result no longer true. Our rst

example is a proof for this fact.

Example 6.1 For the digraph given by Fig. 3,

we have: Mx={x0, y, z},My={y0, x, z},Mz=

{z0, x, y},Mx0={x},My0={y},Mz0={z}

Mx={x0}, My={y0}, Mz={z0},Mx0=

{x0, y, z}, My0={y0, x, z}, Mz0={z0, x, y}.

Example 6.2 The digraph given by Fig. 4 is a

non strongly connected digraph with disconnected

total graphic topology.

Example 6.3 The digraph, given by Fig. 5, is

strongly connected and its connected total graphic

topology.

Example 6.4 In the Fig. 6, the graph is non

strongly connected digraph and its total graphic

topology is disconnected.

In the rest of this paper, we prove some elementary

results about connectedness of the total graphic

topology.

PROOF

DOI: 10.37394/232020.2024.4.3

Hanan Omar Zomam

E-ISSN: 2732-9941

Volume 4, 2024

Figure 4: An example of digraph which is not

strongly connected and its total graphic topology

is disconnected.

Figure 5: This is a strongly connected digraph

with strongly connected total graphic topology.

Figure 6: This is an example of a non strongly con-

nected digraph with a disconnected total graphic

topology.

Theorem 6.1 Let G= (V, E)be a bipartite di-

graph. The total graphic topology Ttot

Gof Gis a

disconnected topology.

Proof. Gis a bipartite graph means there exist

two disjoint subsets B1and B2of Vsuch that

V=B1∪B2and if (x, y)∈Eand (x, y) /∈B1×B2

then (x, y)∈B2×B1.

Consider

U1=[

x∈B1

Mxand U2=[

x∈B2

Mx.

then, U1and U2are nonempty disjoint open

sets of Vsatisfying U1⊂B2,U2⊂B1and

V=U1∪U2. So, (V, Ttot

G)is a disconnected

topological space.

In fact, the last proof conrms the following

result.

Theorem 6.2 Let G= (V, E)be a strongly con-

nected bipartite digraph. The total graphic topol-

ogy Ttot

Gof Gis a disconnected topology.

Theorem 6.3 Suppose that G= (V, E)is a nite

one sense directed cycle, that is, V={x1,· · · , xn}

and E={(xj−1, xj), i = 2 . . . , n}∪{(xn, x1)}.

The total graphic topology Ttot

Gof Gis a discon-

nected topology.

Proof. For all xi∈V,i6= 1, we have Mxi=

{xi−1}and Mx1={xn}. Then, for all x∈V,

{x}is an open set and so Ttot

Gis discrete.

7 Conclusion

In this paper, we introduce the total graphic

topology Ttot

Gon the vertices set of a directed

graph G= (V, E)by using a subbases. The

elements of this subbases are the sets of all

neighbors in any direction of all vertices of the

graph. That is, a neighborhood of a vertex is the

set of all out-neighbors and int-neighbors. For

this reason, the obtained topology are called total

graphic topology. We prove that this topology is

an Alexandro topology that is any intersection

of open set is also an open set. The existing of

minimal bases follows. We give some characteri-

zations of minimal open sets using the subbases.

In addition, we investigate the relation between

isomorphic graphs and their graphic total topolo-

gies and prove that they will be homeomorphic.

The problem of connectedness was investigated

through some examples. As future work, we have

the following question: are there some necessary

and sucient conditions for the connectivity of

(V, Ttot

G)?

PROOF

DOI: 10.37394/232020.2024.4.3

Hanan Omar Zomam

E-ISSN: 2732-9941

Volume 4, 2024

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final findings and solution.

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PROOF

DOI: 10.37394/232020.2024.4.3

Hanan Omar Zomam

E-ISSN: 2732-9941

Volume 4, 2024