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Different Types of Topological Structures by Graphs

Ali Asghar, Ather Qayyum∗, Noor Muhammad
Institute of Southern Punjab, Multan, Pakistan

ali.asghar190289283@gmail.com, atherqayyum@isp.edu.pk, noormustaffa681@gmail.com
∗Correspondence: atherqayyum@isp.edu.pk

Abstract. In this paper, we will represent relation of graph which bring different type of topologicalstructure to the graph [2], then, consider certain properties of the graph. We will discuss mainlyblood circulation in lungs and some different diseases of it [4] and relate them with graph and maketopologies [8]. Moreover, certain applications in medical field will be represent. We can also useresults in real life [11].

1. Introduction and Preliminaries
Initially in eighteenth century swiss mathematician Leonhard Euler gave the basic idea aboutgraph [2]. He resolved famous problems. He drew any tenth spectral graph theory introducedin decade of 1950, while in 1980 introduced monograph spectra by Cvetkovics, Doob and Sachs.Recently graph theory has become very large field not only for mathematicians but also for otherfields of life [13]. In real life graph theory playing its vital role of life, Very common example of it isall roads and motorways form a large network which is used by cruising services e.g. goggle mapswhen working on different routs between two points. Graph theory is the study of graph, whichmathematically used to develop pairwise relationship between objects [13]. This is also a collectionof points and lines. Points are known as vertices and lines are edges. The collection of verticesof any graph G is vertex set and collection of edges is known as edge set denominated as V(G)and E(G) respectively [4]. The number of vertices and edges in G is known as order and size of Grespectively. If an edge has same end is loop. Whenever more than one edges having same finalpoint than it will consider parallel edges [12]. Mapping [14] play a specific role in graph theoryalso.Notions on closure operations are helpful for algebra, topology, basic graph theory and alsofor many other fields [4]. Topology is very advance field of mathematics. It deals with thinsindependently. It allow to increase or decrease things without cutting. Consider [11] X might be
Received: 23 Dec 2021.
Key words and phrases. topological space; graph; relation.1

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Eur. J. Math. Anal. 10.28924/ada/ma.3.3 2nonempty set furthermore φ is collection for X, than (1) φ,X belongs to τ (2) Absolute unionfor number of τ belongs to τ (3) Limited intersection for τ belongs to τ . Than τ will be considertopology over X so, (X, τ) is called topological space. Topology also helpful in different propertieslike Convergence, Existence, Convexity and many other. All elements within topology known asopen set and complement might be close [13].Consider G is any graph, than two adjacent vertices are called nbhd of each other
N (V ) = {u ∈ v (G) | u be nbhd of V }

is open nbhd for V and N[V ] = N (V ) ∪ {V } is closed nbhd for V. [10]Loops and parallel edge free graph is simple graph. If any two distinct vertices joined by anedge is named as complete graph. [12]If vertices of two sets A and B joined by each edge between A and B is called bipartite graph. Ifeach vertex from A connected with every vertices of B with only a single edge is complete bipartitegraph [8].If we delete any edge from a subgraph G is called spanning subgraph while deleting any vertexis induced subgraph [3].Consider that if any subgraph do not contain their final point that channel P will be nominatedby topological open subgraph while having its initial and final point is topological closed graph [7].Consider G = (V, E) be any connected graph. Moreover, (V (G) , τ) be topology [7] generatewith
βj = {V (G) , φ, {Vj}, {N

(
Vj

)
}}

is basis moreover consider S1 and S2 be two open paths than
(I) V (S1) ⊆ Cl V (S1)

(II)S1 ⊆ S2and
Cl (V (S1)) ⊆ Cl (V (S2))

2. Relation Over Graph
Suppose that U is vertex in any graph G having l∗ loop and m multiple edges than

(degG (u))u = (2lu +mu)u

while simple graph is (degG (u))u . [2]
[4] Here is relation R for any graph G is deformed by

R = {((2lu +mu)u , (2lw +mw )w ), u, w ∈ V }

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Eur. J. Math. Anal. 10.28924/ada/ma.3.3 3While lu and lw are number for loops for vertices u ,w from each furthermore mu , mw are multipleedges for vertex u and w respectively. Consider that G be simple graph,
R = {((degG (u)u , degG (w)w )) ; u, w ∈ V }if l = 0 than

R = {(lu)u , (mw )w , u, w ∈ W}if m = 1 and l = 0 than
R = {(lu, lw ) u, w ∈ VConsider G is directed along with simple than

R = {(lu, lw ) = (U,W ) u, w ∈ V }

while if G is undirected than
R = {(lu, lw ) = (u, w)or
(w, u) u, w ∈ V }

Example 1. [4]

Suppose that G is undirected graph given above Fig.1.
R

= {(11a, 8b) , (11a, 5c) , (11a, 8d) , (8b, 5c) , (8b, 8d) ,

(5c , 8d) , (11a, 11a) , (8b, 8b) , (8d , 8d)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.3 4

Example 2. [1]

Let G be a graph given in figure 02
R = {(3c , 3b) , (4a, 5e) , (3b, 3c) , (3c , 3d) , (3c , 5e) , (5e , 3d)}

Example 3. [2]

Let G be a graph in figure 3
R = {(3a, 2b) , (3a, 2d) , (3a, 3c) , (2b, 3c) , (3c , 2d)

3. Topological Structure on Graph
By previous illustration ( 1) created a topology. According to this example the vertices are givenas

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Eur. J. Math. Anal. 10.28924/ada/ma.3.3 5

(11a)R = {8b, 8d , 5c}, (8b)R = {11a, 5c , 8d}, (5c)R = {8b, 8d , 11a}, (8d)R = {11a, 8b, 5c}Subbase
SG

= {{8b, 8d , 5c}, {11a, 5c , 8d}, {8b, 8d , 11a}, {11a, 8b, 5c}}

Topology
τG

= {X,φ, {8b, 8d , 5c}, {11a, 5c , 8d}, {8b, 8d , 11a},

{11a, 8b, 5c}, {8d , 5c}, {8b, 8d}, , {8b, 5c},

{11a, 8d}, {11a, 5c}, {11a, 8b}, {8b, 5c , 8d},

{11a, 5c , 8d}, {11a, 8b, 8d}, {11a, 8b, 5c}

By previous illustration( 2) created a topology. According to this example the vertices are given as
(4a)R

= {3b, 5e}, (3b)R = {4a, 3c}, (5e)R

= {4a, 3c , 3d}, (3c)R = {3b, 3d , 5e}, (3d)R = {3c , 5e}

Subbase
SG

= {{3b, 5e}, {4a, 3c}, {4a, 3c , 3d}, {4a, 3d , 5e}, {3c , 5e}}

Base
βG

= {X,φ, {3b, 5e}, {4a, 3c}, {4a, 3c , 3d}, {3b, 3d , 5e}, {3c , 5e}, {5e}, {3c}, {3d}

Topology
τG

= {X,φ, {3b, 5e}, {4a, 3c}, {4a, 3c , 3d}, {3b, 3d , 5e}, {3c , 5e}, {3c},

{3d}, {5e}, {4a, 3b, 3c , 5e}, {4a, 3c , 3d , 5e}, {3b, 3c , 5e}, {4a, 3c , 5e}, {4a, 3d},

{3b, 3c , 3d , 5e}, {3c , 3d , 5e}, {3c , 3d}, {3c , 5e}, {3d , 5e}

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Eur. J. Math. Anal. 10.28924/ada/ma.3.3 6By previous example (3) it is given as.
(3a)R

= {2b, 3c , 2d}, (2b)R = {3a, 3c}, (3c)R

= {2b, 3a, 2d}, (2d)R = {3a, 3c}

Subbase
SG = {{2b, 3c , 2d}, {3a, 3c}, {2b, 3a, 2d}, {3a, 3c}

Base
βG

= {X,φ, {2b, 3c , 2d}, {3a, 3c}, {3a, 2b, 2d},

{3a, 3c}, {3c}, {2b, 2d}, {3a}

Topology
τG

= {X,φ, {2b, 3c , 2d}, {3a, 3c}, {3a, 2b, 2d},

{3a, 3c}, {3c}, {2b, 2d}, {3a}, {2b, 3c , 2d}

Consider that G = (V ∗, E∗) is graph moreover H is induced subgraph for G. So,
cl (V ∗ (H)) = V (H)U{x ∈ v∗ (G) ; xR ∩ V (H) 6= φ

furthermore
xR = {

(
degG (ar )ar

)
}

∀ r ∈ I and ar is set of every adjacent vertices Vi .Suppose that G = (V ∗, E∗) is graph. Moreover H IS induced subgraph for G and
Int (V ∗ (H)) = {x ∈ V ∗ (G) ; xR ⊆ V (H) , xR = {

(
degG (ar )ar

)
∀ r ∈ I and ar adjacent for X.

4. Some Applications
In this part we will give an example of blood circulation in lungs. We will also draw topologicalstructure of this circulation. We will relate mathematics with medical field. We will made graph ofit. Moreover, we will discuss few reasons of disability in lungs and cause of dangerous diseases.We will explain these diseases mathematically.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.3 7

Here we will utilize our work discussed above in medical field. We will introduced the techniquein which connected graph is modifying condition in the medical field. Diagram represent to graph..

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Eur. J. Math. Anal. 10.28924/ada/ma.3.3 8We can notice the blood circulation in lungs is representation of set of vertices and edges. Than, wecan define a topological structure τG on that. Post classes for vertices in graph are the followinggiven below.
(a1)R = {c3}, (b2)R = {c3}, (c3)R = {d4}, (d4)R = {f5}, (f5)R = {g6, h7}, (g6)R =

{j9}, (h7)R = {i8}, (i8)R = {k10}, (j9)R = {k10}, (k10)R = {p11}, (p11)R = {q12}, (q12)R =
{r13, s14},
(r13)R = {a1}, (s14)R = {b2}The subbase has a form

SG

= {{c3} , {d4} , {f5} , {g6, h7} , {j9} , {i8} , {k10} , {p11} ,

{q12} , {r13,s14} , {a1} , {b2}}

Base has a form
βG

= {X,φ, {c3}, {d4}, {f5}, {g6, h7}, {j9}, {i8},

{k10}, {p11}, {q12}, {r13, s14}, {a1}, {b2}}

Topology on a graph G have
τG

= {X,φ, {c3}, {d4}, {f5}, {g6, h7}, {j9}, {i8}, {k10}, {p11},

{q12}, {r13, s14}, {a1}, {b2},

{c3, d4}, {c3, f5}, {c3, g6, h7}, {c3, j9}, {c3, i8},

{c3, k10}, {c3, p11}, {c3, q12},

{c3, r13, s14}, {c3, a1}, {c3, b2}, {d4, f5},

{d4, g6, h7}, {d4, j9}, {d4, i8}, {d4, k10},

{d4, p11}, {d4, q12}, {d4, r13, s14}, {d4, a1},

{d4, b2}, {f5, g6, h7}, {f5, j9}, {i8}, {f5, k10},

{f5, p11}, {f5, q12}, {f5, r13, s14}, {f5, a1},

{f5, b2}, {g6, h7, j9}, {g6, h7, i8}, {g6, h7, k10},

{g6, h7, p11}, {g6, h7, q12}, {g6, h7, r13, s14},

{g6, h7, a1}, {g6, h7, b2}, {j9, i8}, {j9, k10},

{j9, p11}, {j9, q12}, {j9, r13, s14}, {j9, a1},

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Eur. J. Math. Anal. 10.28924/ada/ma.3.3 9

{j9, b2}, {k10, p11}, {k10, q12}, {k10, r13, s14},

{k10, a1, b2}, {p11, q12}, {p11, r13, s14},

{p11, a1}, {p11, b2}, {q12, r13, s14}, {q12, a1, }

{q12, b2}, {r13, s14, a1}, {r13, s14, b2}, {a1, b2}

Initially we get closure of graph. If H is any subgraph
H = {b2, c3, e2, e3, e4}

that is
V (H) = {b2, c3}by definition of closure for subgraph H be

cl (V (H)) = {b2, c3, d4}

Medically, here we will use that illustration for circulation of blood in lungs will be true. Bloodflow in lungs by directed path to complete its cycle. But due to any fault flow of blood distributeand stop. It create serious diseases. Moreover, we can find interior for graph over subgraph
H = {f5, e5, g6, e7, h7}

but from definition we can assume
intV (H) = {f5, g6}In this example we note that end point does not include. This contradiction in heart but suitablefor lungs medically because due to some disorder people can also survive with only one lungs thisis gift of God.

5. Some Serious Diseases in Lungs
There are some diseases in lungs due to some disorder. These diseases are divided in to somecategories. We will discuss reasons of these diseases and express them graphically. Moreover, wewill also show topological structure. [3]

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Eur. J. Math. Anal. 10.28924/ada/ma.3.3 105.1. Pulmonary Arterial Hypertension. Heart problem autoimmune system can cause high bloodpressure in pulmonary arteries.
(f5)R = {g6, h7}, (g6)R = {i8}, (h7)R = {j9}

Subbase
SG = {{g6, h7}, {i8}, {j9}}Base

βG = {X,φ, {g6, h7}, {i8}, {j9}}Topology
τG

= {X,φ, {g6, h7}, {i8}, {j9}, {g6, h7, i8}, {g6, h7, j9}, {h8, j9}}

5.2. Pulmonary Venous Hypertension. Any damage or self eating of mitral valve can cause higherblood pressure in pulmonary veins.
(i8)R = {k10}, (j9)R = {k10}, (k10)R = {p11}

Subbase
SG = {{k10}, {p11}}Base

βG = {X,φ, {k10}, {p11}}

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Eur. J. Math. Anal. 10.28924/ada/ma.3.3 11Topology
τG = {X,φ, {k10}, {p11}, {k10, p11}}

5.3. Pulmonary Embolism. Any coagulation of blood or fat droplets can travel to lungs from heartand cause blockage of lungs blood vessels.
(c3)R

= {d4}, (d4)R = {f5}, (f5)R = {g6, h7},

(g6)R

= {i8}, (h7)R = {j9}, (i8)R = {k10}, (j9)R = {k10}

Subbase
SG = {{d4}, {f5}, {g6, h7}, {i8}, {j9}, {k10}}

Base
βG = {X,φ, {d4}, {f5}, {g6, h7}, {i8}, {j9}, {k10}, }

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Eur. J. Math. Anal. 10.28924/ada/ma.3.3 12Topology
τG

= {X,φ, {d4}, {f5}, {g6, h7}, {i8}, {j9},

{k10}, {d4, f5}, {d4, g6, h7}, {d4, i8},

{d4, j9}, {d4, k10}, {f5, g6, h7}, {f5, i8},

{f5, j9}, {f5, k10}, {g6, h7, i8}, {g6, h7, j9},

{g6, h7, k10}, {i8, j9}, {i8, k10}, {j9, k10}

6. Conclusion
We derive topological structure by using different relations defined above. We also use differenttype of graphs. We also mentioned the method of general topology its graph and relationshipbetween both of them. We also represent medical field, blood circulation in lungs, its diseasesmade topologies by using graph.

References
[1] G. Chartrand, L. Lesniak, P. Zhang, Textbook in mathematics (Graphs and Diagraphs), Sixth edition, Taylor andFrancis, 2016.[2] M. Shokry, R.E. Aly, Topological properties on graph vs medical application in human heart, Int. J. Appl. Math. 15(2013) 1103-1109.[3] J. Chen, J. Li, An application of rough sets to graph theory, Inform. Sci. 201 (2012) 114–127. https://doi.org/

10.1016/j.ins.2012.03.009.[4] M. Shokray, Y.Y. Yousif, closure operators on graph, Aust. J. Basic Appl. Sci. 5 (2011) 1856-1864.[5] G. Birkhoff, Lattic theory, Amer Math. Soc. 1967.[6] Z. Bonikowski, A representation theorem for co-diagonalizable algebras, Rep Math Logic, 38 (2004) 13-22.[7] D. Dikranjan, W. Tholen, Catogrical structure of closure operator, mathematics and its application, Kluwer AcademicPublisher, Dordrecht, 1995.[8] C. Kuratowski, Topolgies, Warsaw, 1952.[9] W. Shi, K. Liu, A fuzzy topology for computing the interior, boundary, and exterior of spatial objects quantitativelyin GIS, Computers Geosci. 33 (2007) 898–915. https://doi.org/10.1016/j.cageo.2006.10.013.[10] C. Largeron, S. Bonnevay, A pretopological approach for structural analysis, Inform. Sci. 144 (2002) 169–185.
https://doi.org/10.1016/S0020-0255(02)00189-5.[11] B.M.R. Stadler, PF. Stadler, Generalized topological space in involuntarily and combinational chemistry, J. Chem.Inf. Comput. Sci. 42 (2002) 577-585.[12] A. Galton, A generalized topological view of motion in discrete space, Theor. Computer Sci. 305 (2003) 111-134.[13] S.A. Morris, Topology without tears, Online e-book, 2017. https://www.topologywithouttears.net/topbook.
pdf.[14] A. Qayyum, M. Shoaib, M.A. Latif, A generalized inequality of ostrowski type for twice differentiable boundedmappings and applications, Appl. Math. Sci. 8 (2014) 1889-1901.

https://doi.org/10.28924/ada/ma.3.3
https://doi.org/10.1016/j.ins.2012.03.009
https://doi.org/10.1016/j.ins.2012.03.009
https://doi.org/10.1016/j.cageo.2006.10.013
https://doi.org/10.1016/S0020-0255(02)00189-5
https://www.topologywithouttears.net/topbook.pdf
https://www.topologywithouttears.net/topbook.pdf

	1. Introduction and Preliminaries
	2. Relation Over Graph
	3. Topological Structure on Graph
	4. Some Applications
	5. Some Serious Diseases in Lungs
	5.1. Pulmonary Arterial Hypertension
	5.2. Pulmonary Venous Hypertension
	5.3. Pulmonary Embolism

	6. Conclusion
	References

