Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 186 https://internationalpubls.com Structures of Nano-Topologies Via Graphs in the Human Double Circulation System Dr. K. Rekha1, M. Dhanapal2* 1 Assistant Professor, PG and Research Department of Mathematics, Bishop Heber College (Autonomous and Affiliated to Bharathidasan University, Trichy), Tiruchirappalli-620017, Tamilnadu, India. *2Assistant Professor, Department of Mathematics, Kongunadu College of Engineering and Technology (Autonomous), Tiruchirappalli-621215, Tamilnadu, India. *2Corresponding Author Mail Id: mdhanapal46@gmail.com, 1rekhamath84@gmail.com Article History: Received: 17-10-2024 Revised: 01-12-2024 Accepted: 10-12-2024 Abstract: Introduction: The key theme of present research work is to propose an initial left (resp. right) neighbourhood system for an ordering on nano-topological spaces. Based on these neighbourhoods, we explain how the connection between nano-topological spaces and digraph theory may be used in the study of human double circulation system. Objectives: The“most current findings and medical application methods utilize graph theory to demonstrate certain complicated structures with advanced application trends. The basis of this work is the creation of a“nano-topological structures for the supremacy position of vertices of plain”digraphs. We use digraph for derive the nano-topologies in many scientific and medical models to ensure functionality. Methods: To construct a directed graph for the human double circulation system. In addition we derive the nano-topologies by using initial left (resp. right) neighbourhoods for certain probable sub-graphs and also verified the comparison between these two types of neighbourhoods Results: In this paper we have derived certain nano-topological properties by creating initial left (resp. right) neighbourhoods and also we have identified the union and intersection of initial neighbourhoods might help us to ensure the cause of lung and heart failure. Conclusions: The findings of the digraph application approaches that we acquired are really helpful to ensure the functionality of double circulation system in the human body. Keywords: Digraph of human double circulation system, Nano-topological space, initial lower (resp. upper) approximation and initial left (resp. right) neighbourhoods. 1.Introduction Richard and Lellis Thivagar proposed the notion of nano-topology. They developed the theme of Nano-topology is based on a general topology from Pawlak rough set approximations. Graph theory is currently a growing field of study as well as a key to establish the mathematical application for various fields such as statistics, physics, chemistry, sociology and biology. In order to give more generalised nano-topology emerging from graphs, several innovative types of topological structures on a basic directed graph has been derived. A new method for applying nano- Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 187 https://internationalpubls.com topological graph reduction to analysis of symbolic circuits has been built using the graph's structural equivalence in nano-topology. The estimated sub-graphs of graphs based on the neighbourhood have been established and framed using an algorithm designed to detect patent infringement cases. As an example of a practical application, Ashraf S. Nawar and El Atik A.A have used a neighbourhood network of a digraph to generate new types of nano-topological spaces in human heart [1]. It is advantageous to develop a tool to address the human organs' blood circulation system. Creating a nano-topological construction using the power set of vertices of basic digraphs serves as the basis for this effort.We utilise the relationship between digraph theory and nano-topological spaces to address practical issues in various scientific and medical frameworks. The idea behind this study is to present new kinds of nano-topological structures that are generated by graphs via a system of initial vertex neighbourhood systems to the digraph. Certain characteristics of the generated nano-topological spaces, featuring an initial left border area, initial left upper approximation, and initial left lower approximation, have been demonstrated. In the application of this research, we have created the nano-topological structures generated by the same graph with regard to the initial left neighbourhood of the vertex using a simple digraph of the human lung. An assumption of Nano-topology was projected by“LellisThivagar”and Richard. The idea“of Nano- topology is based on a general topology introduced from Pawlak rough set approximations.Graph theory is currently a growing field of study as well as a key to establish the mathematical application for various fields such as fuzzy theory, physics, chemistry, sociology and biology. A“quantity of recent forms of Nano-topological structures on a simple directed graph and provide added generalized Nano-topology induce by graphs are recognized”byArafaNasef,“Abd El Fattah El- Atik”[8].“A new nano-topological graph cutback to representational circuit testing has been urbanized by way of structural equality in Nano-topology induce”bythe”graph.“Analgorithm“have framed for detecting patent infringement suits by LellisThivagar M, Paul Manuel and Sutha Devi”V.They defined the approximation of the sub-graphs. In which“a newest nano-topological graph drop to figurative trip investigation is expanded”[7]. The“most current findings and medical application methods utilize graph theory to demonstrate certain complicated structures with advanced application trends”[2]. The basis of this work is the creation of a“nano-topological structures for the supremacy position of vertices of plain”digraphs.We use digraph for derive the nano-topologies in many scientific and medical models to ensure functionality. 2.Objectives A summary of several fundamental concepts in graph theory and Nano-topology is provided below. Definition:2.1[4] The equivalency relation on U is denoted by R𝑎𝑛𝑑 𝑈 be the universe and 𝜏𝑅(𝐸) = {U, 𝜑, L𝑅 (𝐸), U𝑅 (𝐸),B𝑅 (𝐸)} where 𝐸 ⊆ 𝑈. 𝜏𝑅(𝐸)fulfils the axioms: (i) 𝑈and  𝜑 ∈ 𝜏𝑅(𝐸). (ii) 𝜏𝑅(𝐸)contains the union of all the components of any subcollection. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 188 https://internationalpubls.com (iii)𝜏𝑅(𝐸)contains the intersection of all the components of a finite subcollection. On U, 𝜏𝑅(𝐸)creates a nano-topology that matches E.Through the method of nano-topological space, we refer to {U, 𝜏𝑅(𝐸)}. The term nano-topology refers to the parameters of τR (E). Definition: 2.2[11] Graph G(ρ,σ), where σ=σ(G) is a subset of unordered pairs of ρ and ρ=ρ(G) is nonempty. If G(resp. infinite) is finite, then ρ(G) is also finite (resp. infinite). The number of edges that contain a vertex u in ρ(G) is its degree; if u's degree is zero, then u is referred to as an isolated point. An edge with a vertex that is similar to its ends is called a loop, whereas an edge with specific ends is called a link. Definition: 2.3[2] If u, o∈O and G(ρ,σ) is a directed graph, then: o 𝐼𝑛 𝑐𝑎𝑠𝑒 𝑢𝜌 ∈ 𝜎(𝐺), 𝑙𝑒𝑡 𝑢 𝑏𝑒 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 𝑣𝑒𝑟𝑡𝑒𝑥 𝑜𝑓 ρ. o 𝐼𝑓 𝜌𝑢 ∈ 𝜎(𝐺), 𝑡ℎ𝑒𝑛 𝑢 𝑏𝑒 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑣𝑒𝑟𝑡𝑒𝑥 𝑜𝑓 ρ. o The number of vertices u such that 𝑢𝜌 ∈σ(G) is the left degree of a vertex ρ. o The number of vertices u such that 𝜌𝑢 ∈σ(G) is the correct degree of a vertex ρ. Definition: 2.4[3] If �̂�(�̂�, �̂�) is simple digraph, �̂� ∈ 𝜃(�̂�)then new varieties of initial neighbourhoods of �̂�𝐼 𝑖(�̂�) are computed as: (1) Initial left neighbourhood:�̂�𝑙 𝑖(�̂�) = {�̂� ∈ 𝜃(�̂�)/�̂�𝑙 𝑖(�̂�) ⊆ �̂�𝑙 𝑖(�̂�)} (2) Initial right neighbourhood: �̂�𝑟 𝑖(�̂�) = {�̂� ∈ 𝜃(�̂�)/�̂�𝑟 𝑖(�̂�) ⊆ �̂�𝑟 𝑖(�̂�)} (3) Union of initial neighbourhoods: �̂�𝑢 𝑖 (�̂�) = �̂�𝑙 𝑖(�̂�) ∪ �̂�𝑟 𝑖(�̂�) (4) Intersection of initial neighbourhoods:�̂�𝑖 𝑖(�̂�) = �̂�𝑙 𝑖(�̂�) ∩ �̂�𝑟 𝑖(�̂�) Definition: 2.5 If �̂�be sub-graph of �̂�, �̂�𝑙 𝑖(�̂�)be the initial leftneighbourhoodsof�̂� ∈ 𝜃(�̂�)and�̂�(𝜃, �̂�) be a simple digraph,at that point, (i) Initial left lower-approximationis 𝐿𝑁𝑙 𝐼[𝜃(�̂�)] = ⋃ {�̂�/𝑁𝑙 𝑖(�̂�) ⊆ 𝜃(�̂�)}�̂�∈�̂�(�̂�) (ii) Initial left upper-approximationis 𝐿𝑁𝑙 𝐼[𝜃(�̂�)] = ⋃ {�̂�/𝑁𝑙 𝑖(�̂�) ∩ 𝜃(�̂�) ≠ 𝜑}�̂�∈�̂�(�̂�) (iii) Initial left boundary region is �̂�𝑁𝑙 𝐼[𝜃(�̂�)] = 𝐿𝑁𝑙 𝐼[𝜃(�̂�)] − 𝐿𝑁𝑙 𝐼[𝜃(�̂�)] That is�̂� 𝑁𝑙 𝑖[𝜃(�̂�)] = {𝜃(�̂�), 𝜑, 𝐿𝑁𝑙 𝐼[𝜃(�̂�)], 𝐿𝑁𝑙 𝐼[𝜃(�̂�)], �̂�𝑁𝑙 𝐼[𝜃(�̂�)]}describes the topology of a graph�̂�(𝜃, �̂�)captioned "Graph's enticing nanotopology" with corresponds to a vertex 𝑁𝑙 𝑖(𝜌). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 189 https://internationalpubls.com Definition: 2.6[5] Let �̂�(𝜃, �̂�) is simple digraph, �̂� be a sub-graph of �̂�. Then𝑁𝑙 𝑖(�̂�) is initial left neighbourhood and �̂�𝑅(𝑋) = {𝜃(�̂�), 𝜑, 𝐿𝑁𝑙 𝐼[𝜃(�̂�)], 𝐿𝑁𝑙 𝐼[𝜃(�̂�)], �̂�𝑁𝑙 𝐼[𝜃(�̂�)]}. That is �̂�𝑅(𝜎) forms a nano-topology on�̂�(𝜃, �̂�)corresponding to neighborhood of vertex 𝑁𝑙 𝑖(�̂�). We call (𝜃(�̂�), �̂�𝑁𝑙 𝐼[𝜃(�̂�)])as nano- topological space encouraged by initial left neighbourhood. Definition: 2.7 If �̂� be sub-graph of �̂�, �̂�𝑟 𝑖(�̂�) be the initial right neighbourhoods of �̂� ∈ 𝜃(�̂�) 𝑎𝑛𝑑 �̂�(𝜃, �̂�) be a digraph,at that point, (i) Initial right lower-approximation is 𝐿𝑁𝑟 𝐼 [𝜃(�̂�)] = ⋃ {�̂�/𝑁𝑟 𝑖(�̂�) ⊆ 𝜃(�̂�)}�̂�∈�̂�(�̂�) (ii) Initial right upper-approximation is 𝐿𝑁𝑟 𝐼 [𝜃(�̂�)] = ⋃ {�̂�/𝑁𝑟 𝑖(�̂�) ∩ 𝜃(�̂�) ≠ 𝜑}�̂�∈�̂�(�̂�) (iii) Initial right boundary region is �̂�𝑁𝑟 𝐼 [𝜃(�̂�)] = 𝐿𝑁𝑟 𝐼 [𝜃(�̂�)] − 𝐿𝑁𝑟 𝐼 [𝜃(�̂�)] That is�̂�𝑁𝑟 𝑖 [𝜃(�̂�)] = {𝜃(�̂�), 𝜑, 𝐿𝑁𝑟 𝐼 [𝜃(�̂�)], 𝐿𝑁𝑟 𝐼 [𝜃(�̂�)], �̂�𝑁𝑟 𝐼 [𝜃(�̂�)]}describes the topology of a graph�̂�(𝜃, �̂�)captioned "Graph's enticing nanotopology" with corresponds to a vertex 𝑁𝑟 𝑖(𝜌). 3.Methods SOME NANO-TOPOLOGIES BY USINGINITIAL NEIGHBOURHOODS Example: 3.1 Consider a simple directed graph �̂�with 𝜃(�̂�) = {�̂�1,  𝜌2̂,  �̂�3,  �̂�4} and 𝜃(�̂�) ⊆ 𝜃(�̂�). Figure 1: A simple directed graph�̂� Next,we obtain, Table 3.1: Initial left and right neighborhoods for simple directed graph Ĝ �̂� ∈ 𝜃(�̂�) �̂�1 �̂�2 �̂�3 �̂�4 𝑵𝒍 (�̂�) {�̂�1, �̂�3} {�̂�1, �̂�2, �̂�4} {�̂�2, �̂�3, �̂�4} {�̂�1, �̂�4} 𝑵𝒍 𝒊(�̂�) {�̂�1} {�̂�2} {�̂�3} {�̂�2, �̂�4} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 190 https://internationalpubls.com 𝑵𝒓 (�̂�) {�̂�1, �̂�2, �̂�4} {�̂�2, �̂�3} {�̂�1, �̂�3} {�̂�2, �̂�3, �̂�4} 𝑵𝒓 𝒊 (�̂�) {�̂�1} {�̂�2, �̂�4} {�̂�3} {�̂�4} Table 3.2: Nano-topologies for possible subgraphs of Ĝ by using initial leftneighborhoods. Table 3.3: Nano-topologies for possible subgraphs of G by using initial right neighborhoods. �̂�(�̂�) 𝑳𝑵𝒍 𝒊[�̂�(�̂�)] 𝑳𝑵𝒍 𝒊[�̂�(�̂�)] �̂�𝑵𝒍 𝒊[�̂�(�̂�)] �̂�𝑵𝒍 𝒊[�̂�(�̂�)] {�̂�1} {�̂�1} {�̂�1} 𝜑 {𝜃(�̂�), 𝜑, {�̂�1}} {�̂�2} {�̂�2} {�̂�2} 𝜑 {𝜃(�̂�), 𝜑, {�̂�2}} {�̂�3} {�̂�3} {�̂�3} 𝜑 {𝜃(�̂�), 𝜑, {�̂�3}} {�̂�4} 𝜑 {�̂�2, �̂�4} {�̂�2, �̂�4} {𝜃(�̂�), 𝜑, {�̂�2, �̂�4}} {�̂�1, �̂�2} {�̂�1, �̂�2} {�̂�1, �̂�2} 𝜑 {𝜃(�̂�), 𝜑, {�̂�1, �̂�2}} {�̂�1, �̂�3} {�̂�1, �̂�3} {�̂�1, �̂�3} 𝜑 {𝜃(�̂�), 𝜑, {�̂�1, �̂�3}} {�̂�1, �̂�4} {�̂�1} {{�̂�1, �̂�2 , �̂�4}} {�̂�2 , �̂�4} {𝜃(�̂�), 𝜑, {�̂�1}, {{�̂�1, �̂�2 , �̂�4}}, {�̂�2, �̂�4}} {�̂�2, �̂�3} {�̂�2, �̂�3} {�̂�2, �̂�3} 𝜑 {𝜃(�̂�), 𝜑, {�̂�2, �̂�3}} {�̂�2, �̂�4} {�̂�2, �̂�4} {�̂�2, �̂�4} 𝜑 {𝜃(�̂�), 𝜑, {�̂�2, �̂�4}} {�̂�3, �̂�4} {�̂�3} {�̂�2, �̂�3, �̂�4} {�̂�2 , �̂�4} {𝜃(�̂�), 𝜑, {�̂�3}, {�̂�2, �̂�3, �̂�4}, {�̂�2 , �̂�4}} {�̂�1, �̂�2 , �̂�3} {�̂�1, �̂�2 , �̂�3} {�̂�1, �̂�2 , �̂�3} 𝜑 {𝜃(�̂�), 𝜑, {�̂�1, �̂�2 , �̂�3}} {�̂�1, �̂�2 , �̂�4} {�̂�1, �̂�2 , �̂�4} {�̂�1, �̂�2 , �̂�4} 𝜑 {𝜃(�̂�), 𝜑, {�̂�1, �̂�2 , �̂�4}} {�̂�1, �̂�3 , �̂�4} {�̂�1, �̂�3} V ( Ĝ ) {�̂�2, �̂�4} {𝜃(�̂�), 𝜑, {�̂�1, �̂�3}, {�̂�2, �̂�4}} {�̂�2, �̂�3, �̂�4} {�̂�2, �̂�3, �̂�4} {�̂�2, �̂�3, �̂�4} 𝜑 {𝜃(�̂�), 𝜑, {�̂�2, �̂�3, �̂�4}} 𝜃(�̂�) 𝜃(�̂�) 𝜃(�̂�) 𝜑 {𝜃(�̂�), 𝜑} 𝜑 𝜑 𝜑 𝜑 {𝜃(�̂�), 𝜑} 𝑉 ( Ĥ ) 𝐿𝑁𝑟 ′ [𝑉 ( Ĥ )] 𝑈𝑁𝑟 ′ [𝑉 ( Ĥ )] 𝐵𝑁𝑟 ′ [𝑉 ( Ĥ )] 𝜏𝑁𝑟 ′ [𝑉 ( Ĥ )] {�̂�1} {�̂�1} {�̂�1} 𝜑 {𝜃(�̂�), 𝜑, {�̂�1}} {�̂�2} 𝜑 {�̂�2, �̂�4} {�̂�2, �̂�4} {𝜃(�̂�), 𝜑, {�̂�2, �̂�4}} {�̂�3} {�̂�3} {�̂�3} 𝜑 {𝜃(�̂�), 𝜑, {�̂�3}} {�̂�4} {�̂�4} {�̂�4} 𝜑 {𝜃(�̂�), 𝜑, {�̂�4}} {�̂�1, �̂�2} {�̂�1} {�̂�1, �̂�2 , �̂�4} {�̂�2, �̂�4} {𝜃(�̂�), 𝜑, {�̂�2, �̂�4}, {�̂�1, �̂�2 , �̂�4}} {�̂�1, �̂�3} {�̂�1, �̂�3} {�̂�1, �̂�3} 𝜑 {𝜃(�̂�), 𝜑, {�̂�1, �̂�3}} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 191 https://internationalpubls.com Observation: 3.2 If �̂�(𝜃, �̂�)be a simple digraph, �̂� and �̂�are two subgraphs of Ĝ then by previous example 3.2, we state some properties as follows: (i) 𝐿 𝑁𝑙 𝑖[𝜃(�̂�) ∪ 𝜃(�̂�)] ⊆ 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] ∪ 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] (ii) 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)𝜃(�̂�)]𝐿 𝑁𝑙 𝑖[𝜃(�̂�)]𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] (iii) 𝐿𝑁𝑟 𝑖 [𝜃(�̂�) ∪ 𝜃(�̂�)] ⊆ 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] ∪ 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] (iv) 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)𝜃(�̂�)]𝐿𝑁𝑟 𝑖 [𝜃(�̂�)]𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] Proof:(i)𝐿 𝑁𝑙 𝑖[𝜃(�̂�) ∪ 𝜃(�̂�)] ⊆ 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] ∪ 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] Let𝜃(�̂�) = {�̂�1, �̂�2 , �̂�4}and 𝜃(�̂�) = {�̂�2, �̂�3, �̂�4} Then𝜃(�̂�) ∪ 𝜃(�̂�) = {�̂�1, �̂�2 , �̂�3, �̂�4} .,.ei 𝐿 𝑁𝑙 𝑖[𝜃(�̂�) ∪ 𝜃(�̂�)] = {�̂�1, �̂�2 , �̂�3, �̂�4}---------------(1) Now .,.ei 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] = {�̂�1, �̂�2 , �̂�4}and𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] = {�̂�2, �̂�3, �̂�4} .,.ei 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] ∪ 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] = {�̂�1, �̂�2 , �̂�3, �̂�4}---------------(2) From equations (1) and (2), we get 𝐿 𝑁𝑙 𝑖[𝜃(�̂�) ∪ 𝜃(�̂�)] ⊆ 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] ∪ 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] (ii)𝐿 𝑁𝑙 𝑖[𝜃(�̂�)𝜃(�̂�)]𝐿 𝑁𝑙 𝑖[𝜃(�̂�)]𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] Let𝜃(�̂�) = {�̂�1, �̂�2 , �̂�4}and𝜃(�̂�) = {�̂�2, �̂�3, �̂�4} {�̂�1, �̂�4} {�̂�1, �̂�4} {�̂�1, �̂�4} 𝜑 {𝜃(�̂�), 𝜑, {�̂�1, �̂�4}} {�̂�2, �̂�3} {�̂�3} {�̂�2, �̂�3, �̂�4} {�̂�2, �̂�4} {𝜃(�̂�), 𝜑, {�̂�3}, {�̂�2, �̂�4}, {�̂�2, �̂�3, �̂�4}} {�̂�2, �̂�4} {�̂�2, �̂�4} {�̂�2, �̂�4} 𝜑 {𝜃(�̂�), 𝜑, {�̂�2, �̂�4}} {�̂�3, �̂�4} {�̂�3, �̂�4} {�̂�3, �̂�4} 𝜑 {𝜃(�̂�), 𝜑, {�̂�3, �̂�4}} {�̂�1, �̂�2 , �̂�3} {�̂�1, �̂�3} V ( Ĝ ) {�̂�2, �̂�4} {𝜃(�̂�), 𝜑, {�̂�1, �̂�3}, {�̂�2, �̂�4}} {�̂�1, �̂�2 , �̂�4} {�̂�1, �̂�2 , �̂�4} {�̂�1, �̂�2 , �̂�4} 𝜑 {𝜃(�̂�), 𝜑, {�̂�1, �̂�2 , �̂�4}} {�̂�1, �̂�3 , �̂�4} {�̂�1, �̂�3 , �̂�4} {�̂�1, �̂�3 , �̂�4} 𝜑 {𝜃(�̂�), 𝜑, {�̂�1, �̂�3 , �̂�4}} {�̂�2, �̂�3, �̂�4} {�̂�2, �̂�3, �̂�4} {�̂�2, �̂�3, �̂�4} 𝜑 {𝜃(�̂�), 𝜑, {�̂�2, �̂�3, �̂�4}} 𝜃(�̂�) 𝜃(�̂�) 𝜃(�̂�) 𝜑 {𝜃(�̂�), 𝜑} 𝜑 𝜑 𝜑 𝜑 {𝜃(�̂�), 𝜑} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 192 https://internationalpubls.com Then 𝜃(�̂�) ∪ 𝜃(�̂�) = {�̂�1, �̂�2 , �̂�3, �̂�4} .,.ei 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)𝜃(�̂�)] = {�̂�1, �̂�2 , �̂�3, �̂�4}---------------(1) Now𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] = {�̂�1, �̂�2 , �̂�4}and𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] = {�̂�2, �̂�3, �̂�4} .,.ei 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)]𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] = {�̂�2, �̂�4}---------------(2) From equations (1) and (2), we get 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)𝜃(�̂�)]𝐿 𝑁𝑙 𝑖[𝜃(�̂�)]𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] (iii)𝐿𝑁𝑟 𝑖 [𝜃(�̂�) ∪ 𝜃(�̂�)] ⊆ 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] ∪ 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] Let𝜃(�̂�) = {�̂�1, �̂�2 , �̂�4}and 𝜃(�̂�) = {�̂�2, �̂�3, �̂�4} Then𝜃(�̂�) ∪ 𝜃(�̂�) = {�̂�1, �̂�2 , �̂�3, �̂�4} .,.ei 𝐿𝑁𝑟 𝑖 [𝜃(�̂�) ∪ 𝜃(�̂�)] = {�̂�1, �̂�2 , �̂�3, �̂�4}---------------(1) Now 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] = {�̂�1, �̂�2 , �̂�4}and 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] = {�̂�2, �̂�3, �̂�4} .,.ei 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] ∪ 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] = {�̂�1, �̂�2 , �̂�3, �̂�4}---------------(2) From equations (1) and (2), we get 𝐿𝑁𝑟 𝑖 [𝜃(�̂�) ∪ 𝜃(�̂�)] ⊆ 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] ∪ 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] (iv)𝐿𝑁𝑟 𝑖 [𝜃(�̂�)𝜃(�̂�)]𝐿𝑁𝑟 𝑖 [𝜃(�̂�)]𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] Let𝜃(�̂�) = {�̂�1, �̂�2 , �̂�4}and𝜃(�̂�) = {�̂�2, �̂�3, �̂�4} Then 𝜃(�̂�) ∪ 𝜃(�̂�) = {�̂�1, �̂�2 , �̂�3, �̂�4} .,.ei 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)𝜃(�̂�)] = {�̂�1, �̂�2 , �̂�3, �̂�4}---------------(1) Now𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] = {�̂�1, �̂�2 , �̂�4}and𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] = {�̂�2, �̂�3, �̂�4} .,.ei 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)]𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] = {�̂�2, �̂�4}---------------(2) From equations (1) and (2), we get 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)𝜃(�̂�)]𝐿𝑁𝑟 𝑖 [𝜃(�̂�)]𝐿𝑁𝑟 𝑖 [𝜃(�̂�)].Hence the proof. Theorem: 3.3 If �̂�(𝜃, �̂�)is a simple digraph and �̂�be a sub graph of �̂� then the followingstatements are holds good: (i) 𝐿 𝑁𝑙 𝑖 {𝐿 𝑁𝑙 𝑖[𝜃(�̂�)]} ⊆ 𝜃(�̂�) ⊆ 𝐿 𝑁𝑙 𝑖 {𝐿 𝑁𝑙 𝑖[𝜃(�̂�)]}    )]ˆ(ˆ[ˆˆ)ˆ(ˆ)]ˆ(ˆ[ˆˆ HVULHVHVLU i l i l i l i l NNNN  (ii) 𝐿𝑁𝑟 𝑖 {𝐿𝑁𝑟 𝑖 [𝜃(�̂�)]} ⊆ 𝜃(�̂�) ⊆ 𝐿𝑁𝑟 𝑖 {𝐿𝑁𝑟 𝑖 [𝜃(�̂�)]}    )]ˆ(ˆ[ˆˆ)ˆ(ˆ)]ˆ(ˆ[ˆˆ HVULHVHVLU i r i r i r i r NNNN  Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 193 https://internationalpubls.com Proof: Proof of (i): Let�̂� ∈ 𝐿 𝑁𝑙 𝑖 {𝐿 𝑁𝑙 𝑖[𝜃(�̂�)]}-----------(i)  �̂� ∈ 𝐿 𝑁𝑙 𝑖{[𝜃(�̂�)]}such that𝑢𝜌 ∈ 𝜎(𝐺) 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] = ⋃ {�̂�/𝑁𝑙 𝑖(�̂�) ⊆ 𝜃(�̂�)} �̂�∈�̂�(�̂�) .,.ei �̂� ∈ 𝑁𝑙 𝑖(�̂�) ⊆ 𝜃(�̂�) �̂� ∈ 𝜃(�̂�)----------- (ii) From equations (i) and (ii), we get 𝐿 𝑁𝑙 𝑖 {𝐿 𝑁𝑙 𝑖[𝜃(�̂�)]} ⊆ 𝜃(�̂�)---------------(1) Now let�̂� ∈ 𝜃(�̂�)----------- (iii) Then𝑁𝑙 𝑖(�̂�) ⊆ 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] Since�̂� ∈ 𝜃(�̂�)and𝑁𝑙 𝑖(�̂�) ⊆ 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] .,.ei �̂� ∈ 𝐿 𝑁𝑙 𝑖 {𝐿 𝑁𝑙 𝑖[𝜃(�̂�)]}-----------(iv) From equations (iii) and (iv), we get 𝜃(�̂�)𝐿 𝑁𝑙 𝑖 ⊆ {𝐿 𝑁𝑙 𝑖[𝜃(�̂�)]}---------------(2) From equations (1) and (2), we get 𝐿 𝑁𝑙 𝑖 {𝐿 𝑁𝑙 𝑖[𝜃(�̂�)]} ⊆ 𝜃(�̂�) ⊆ 𝐿 𝑁𝑙 𝑖 {𝐿 𝑁𝑙 𝑖[𝜃(�̂�)]} Proof of (ii): Let�̂� ∈ 𝐿𝑁𝑟 𝑖 {𝐿𝑁𝑟 𝑖 [𝜃(�̂�)]}----------- (i)  �̂� ∈ 𝐿𝑁𝑟 𝑖 {[𝜃(�̂�)]}such that 𝜌𝑢 ∈ 𝜎(𝐺) 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] = ⋃ {�̂�/𝑁𝑟 𝑖(�̂�) ⊆ 𝜃(�̂�)} �̂�∈�̂�(�̂�) .,.ei �̂� ∈ 𝑁𝑟 𝑖(�̂�) ⊆ 𝜃(�̂�) �̂� ∈ 𝜃(�̂�)----------- (ii) From equations (i) and (ii), we get 𝐿𝑁𝑟 𝑖 {𝐿𝑁𝑟 𝑖 [𝜃(�̂�)]} ⊆ 𝜃(�̂�)---------------(1) Now let �̂� ∈ 𝜃(�̂�)----------- (iii) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 194 https://internationalpubls.com Then𝑁𝑟 𝑖(�̂�) ⊆ 𝐿 𝑁𝑙 𝑖[𝜃(�̂�)] Since�̂� ∈ 𝜃(�̂�)and𝑁𝑟 𝑖(�̂�) ⊆ 𝐿𝑁𝑟 𝑖 [𝜃(�̂�)] .,.ei �̂� ∈ 𝐿𝑁𝑟 𝑖 {𝐿𝑁𝑟 𝑖 [𝜃(�̂�)]}-----------(iv) From equations (iii) and (iv), we get 𝜃(�̂�) ⊆ 𝐿𝑁𝑟 𝑖 {𝐿𝑁𝑟 𝑖 [𝜃(�̂�)]}---------------(2) From equations (1) and (2), we get 𝐿𝑁𝑟 𝑖 {𝐿𝑁𝑟 𝑖 [𝜃(�̂�)]} ⊆ 𝜃(�̂�) ⊆ 𝐿𝑁𝑟 𝑖 {𝐿𝑁𝑟 𝑖 [𝜃(�̂�)]}. Hence the proof. 4.Results NANO-TOPOLOGICAL STRUCTURES VIA GRAPH FOR HUMAN DOUBLE CIRCULATION SYSTEM `The circulatory system is responsible for the transportation of nutrients and gases like oxygen, for the body and metabolic waste products away from the body. The heart and the lungs play an important role in circulating and purification of blood throughout the body. Here we shows that different parts of human double circulation systemin figure-4.1, 4.2 and the corresponding graphical representation as in figure-4.3. DOUBLE CIRCULATION (Figure-4.1) 195 https://internationalpubls.com (Figure-4.2) Let Ĝ(θ̂, σ̂) be a simple digraph (figure-3), L̂ be a sub-graph of Ĝ and the set of vertices of a simple digraph is θ̂(Ĝ) = {ρ̂1,   ρ̂2,  ρ̂3,  ρ̂4, ρ̂5, ρ̂6, ρ̂7, ρ̂8, ρ̂9, ρ̂10, ρ̂11, ρ̂12, ρ̂13, ρ̂14} Where ρ̂1=right atrium,ρ̂2= right ventricle, ρ̂3=pulmonary artery, ρ̂4=arteriols, ρ̂5=lung capillaries, ρ̂6= venules, ρ̂7= pulmonary veins, ρ̂8= left atrium, ρ̂9= left ventricle,ρ̂10=aorta,ρ̂11= arteries and arterioles ,ρ̂12= systemic capillaries,ρ̂13= veins and venules ,ρ̂14= venae cavae. In this wok we find the initial left (resp. right) neighbourhoods of a vertex which can be used as a tool for nano-topological space. Table: 4.1 Left neighbourhood Initial left neighbourhood �̂�𝑙(ρ̂1) = {ρ̂1,   ρ̂14} �̂�𝑙 𝑖(ρ̂1) = {ρ̂1} �̂�𝑙(ρ̂2) = {ρ̂1,  ρ̂2} �̂�𝑙 𝑖(ρ̂2) = {ρ̂2} �̂�𝑙(ρ̂3) = {ρ̂2, ρ̂3} �̂�𝑙 𝑖(ρ̂3) = {ρ̂3} �̂�𝑙(ρ̂4) = {ρ̂3, ρ̂4} �̂�𝑙 𝑖(ρ̂4) = {ρ̂4} �̂�𝑙(ρ̂5) = {ρ̂4, ρ̂5} �̂�𝑙 𝑖(ρ̂5) = {ρ̂5} �̂�𝑙(ρ̂6) = {ρ̂5, ρ̂6} �̂�𝑙 𝑖(ρ̂6) = {ρ̂6} �̂�𝑙(ρ̂7) = {ρ̂6, ρ̂7} �̂�𝑙 𝑖(ρ̂7) = {ρ̂7} �̂�𝑙(ρ̂8) = {ρ̂7, ρ̂8} �̂�𝑙 𝑖(ρ̂8) = {ρ̂8} �̂�𝑙(ρ̂9) = {ρ̂8 , ρ̂9} �̂�𝑙 𝑖(ρ̂9) = {ρ̂9} �̂�𝑙(ρ̂10) = {ρ̂9 , ρ̂10} �̂�𝑙 𝑖(ρ̂10) = {ρ̂10} �̂�𝑙(ρ̂11) = {ρ̂10 , ρ̂11} �̂�𝑙 𝑖(ρ̂11) = {ρ̂11} �̂�𝑙(ρ̂12) = {ρ̂11, ρ̂12} �̂�𝑙 𝑖(ρ̂12) = {ρ̂12} �̂�𝑙(ρ̂13) = {ρ̂12, ρ̂13} �̂�𝑙 𝑖(ρ̂13) = {ρ̂13} �̂�𝑙(ρ̂14) = {ρ̂13, ρ̂14} �̂�𝑙 𝑖(ρ̂14) = {ρ̂14} 196 https://internationalpubls.com Table: 4.2 Right neighbourhood Initial right neighbourhood �̂�𝑟(ρ̂1) = {ρ̂1,  ρ̂2} �̂�𝑟 𝑖(ρ̂1) = {ρ̂1} �̂�𝑟(ρ̂2) = {ρ̂2, ρ̂3} �̂�𝑟 𝑖(ρ̂2) = {ρ̂2} �̂�𝑟(ρ̂3) = {ρ̂3, ρ̂4} �̂�𝑟 𝑖(ρ̂3) = {ρ̂3} �̂�𝑟(ρ̂4) = {ρ̂4, ρ̂5} �̂�𝑟 𝑖(ρ̂4) = {ρ̂4} �̂�𝑟(ρ̂5) = {ρ̂5, ρ̂6} �̂�𝑟 𝑖(ρ̂5) = {ρ̂5} �̂�𝑟(ρ̂6) = {ρ̂6, ρ̂7} �̂�𝑟 𝑖(ρ̂6) = {ρ̂6} �̂�𝑟(ρ̂7) = {ρ̂7, ρ̂8} �̂�𝑟 𝑖(ρ̂7) = {ρ̂7} �̂�𝑟(ρ̂8) = {ρ̂8 , ρ̂9} �̂�𝑟 𝑖(ρ̂8) = {ρ̂8} �̂�𝑟(ρ̂9) = {ρ̂9 , ρ̂10} �̂�𝑟 𝑖(ρ̂9) = {ρ̂9} �̂�𝑟(ρ̂10) = {ρ̂10 , ρ̂11} �̂�𝑟 𝑖(ρ̂10) = {ρ̂10} �̂�𝑟(ρ̂11) = {ρ̂11, ρ̂12} �̂�𝑟 𝑖(ρ̂11) = {ρ̂11} �̂�𝑟(ρ̂12) = {ρ̂12, ρ̂13} �̂�𝑟 𝑖(ρ̂12) = {ρ̂12} �̂�𝑟(ρ̂13) = {ρ̂13, ρ̂14} �̂�𝑟 𝑖(ρ̂13) = {ρ̂13} �̂�𝑟(ρ̂14) = {ρ̂1,  ρ̂14} �̂�𝑟 𝑖(ρ̂14) = {ρ̂14} (Figure-4.3) Table: 4.3 Union of initial Neighbourhoods Intersection of initial neighbourhoods �̂�𝑢 𝑖 (ρ̂1) = {ρ̂1} �̂�𝑖 𝑖(ρ̂1) = {ρ̂1} �̂�𝑢 𝑖 (ρ̂2) = {ρ̂2} �̂�𝑖 𝑖(ρ̂2) = {ρ̂2} �̂�𝑢 𝑖 (ρ̂3) = {ρ̂3} �̂�𝑖 𝑖(ρ̂3) = {ρ̂3} �̂�𝑢 𝑖 (ρ̂4) = {ρ̂4} �̂�𝑖 𝑖(ρ̂4) = {ρ̂4} �̂�𝑢 𝑖 (ρ̂5) = {ρ̂5} �̂�𝑖 𝑖(ρ̂5) = {ρ̂5} �̂�𝑢 𝑖 (ρ̂6) = {ρ̂6} �̂�𝑖 𝑖(ρ̂6) = {ρ̂6} �̂�𝑢 𝑖 (ρ̂7) = {ρ̂7} �̂�𝑖 𝑖(ρ̂7) = {ρ̂7} �̂�𝑢 𝑖 (ρ̂8) = {ρ̂8} �̂�𝑖 𝑖(ρ̂8) = {ρ̂8} �̂�𝑢 𝑖 (ρ̂9) = {ρ̂9} �̂�𝑖 𝑖(ρ̂9) = {ρ̂9} �̂�𝑢 𝑖 (ρ̂10) = {ρ̂10} �̂�𝑖 𝑖(ρ̂10) = {ρ̂10} 197 https://internationalpubls.com �̂�𝑢 𝑖 (ρ̂11) = {ρ̂11} �̂�𝑖 𝑖(ρ̂11) = {ρ̂11} �̂�𝑢 𝑖 (ρ̂12) = {ρ̂12} �̂�𝑖 𝑖(ρ̂12) = {ρ̂12} �̂�𝑢 𝑖 (ρ̂13) = {ρ̂13} �̂�𝑖 𝑖(ρ̂13) = {ρ̂13} �̂�𝑢 𝑖 (ρ̂14) = {ρ̂14} �̂�𝑖 𝑖(ρ̂14) = {ρ̂14} Table 4.4:Nano topologies�̂�𝑵𝒍 𝒊[�̂�(�̂�)]by using initial left neighbourhoods for subgraph of Figure 4.3 �̂�(�̂�) �̂�𝑵𝒍 𝒊[�̂�(�̂�)] {ρ̂1,  ρ̂5} {𝜃(�̂�), 𝜑, {ρ̂1,  ρ̂5}} {ρ̂6,  ρ̂9} {𝜃(�̂�), 𝜑, {ρ̂6,  ρ̂9}} {ρ̂10,  ρ̂13} {𝜃(�̂�), 𝜑, {ρ̂10,  ρ̂13}} {ρ̂2,  ρ̂8} {𝜃(�̂�), 𝜑, {ρ̂2,  ρ̂8}} {ρ̂7,   ρ̂12} {𝜃(�̂�), 𝜑, {ρ̂7,   ρ̂12}} {ρ̂3, ρ̂6,   ρ̂11} {𝜃(�̂�), 𝜑, {ρ̂3, ρ̂6,   ρ̂11}} {ρ̂4, ρ̂6,   ρ̂12} {𝜃(�̂�), 𝜑, {ρ̂4, ρ̂6,   ρ̂12}} {ρ̂5, ρ̂7,   ρ̂13} {𝜃(�̂�), 𝜑, {ρ̂5, ρ̂7,   ρ̂13}} {ρ̂9, ρ̂12,  ρ̂14} {𝜃(�̂�), 𝜑, {ρ̂9, ρ̂12,  ρ̂14}} {ρ̂2, ρ̂3, ρ̂7, ρ̂8,   ρ̂9} {𝜃(�̂�), 𝜑, {ρ̂2, ρ̂3, ρ̂7, ρ̂8,   ρ̂9}} {ρ̂4, ρ̂8,  ρ̂9,   ρ̂11, ρ̂12} {𝜃(�̂�), 𝜑, {ρ̂4, ρ̂8,  ρ̂9,  ρ̂11, ρ̂12}} { ρ̂8,  ρ̂9, ρ̂10,  ρ̂11, ρ̂13} {𝜃(�̂�), 𝜑, { ρ̂8,  ρ̂9, ρ̂10,  ρ̂11, ρ̂13}} {ρ̂2, ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂9} {𝜃(�̂�), 𝜑, {ρ̂2, ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂9}} {ρ̂2, ρ̂4, ρ̂6, ρ̂7, ρ̂9, ρ̂10} {𝜃(�̂�), 𝜑, {ρ̂2, ρ̂4, ρ̂6, ρ̂7, ρ̂9, ρ̂10}} {ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂8, ρ̂9} {𝜃(�̂�), 𝜑, {ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂8, ρ̂9}} {ρ̂5, ρ̂6, ρ̂8,   ρ̂9,   ρ̂12, ρ̂14} {𝜃(�̂�), 𝜑, {ρ̂5, ρ̂6, ρ̂8,   ρ̂9,   ρ̂12, ρ̂14}} {ρ̂1, ρ̂2, ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂9} {𝜃(�̂�), 𝜑, {ρ̂1, ρ̂2, ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂9}} {ρ̂2, ρ̂3, ρ̂4, ρ̂6, ρ̂8, ρ̂9, ρ̂10} {𝜃(�̂�), 𝜑, {ρ̂2, ρ̂3, ρ̂4, ρ̂6, ρ̂8, ρ̂9, ρ̂10}} {ρ̂4, ρ̂5, ρ̂6, ρ̂8, ρ̂9, ρ̂10, ρ̂11} {𝜃(�̂�), 𝜑, {ρ̂4, ρ̂5, ρ̂6, ρ̂8, ρ̂9, ρ̂10, ρ̂11}} {ρ̂3, ρ̂5, ρ̂6, ρ̂9, ρ̂10, ρ̂11, ρ̂12} {𝜃(�̂�), 𝜑, {ρ̂3, ρ̂5, ρ̂6, ρ̂9, ρ̂10, ρ̂11, ρ̂12}} {ρ̂5, ρ̂7, ρ̂8, ρ̂9, ρ̂10, ρ̂11, ρ̂13} {𝜃(�̂�), 𝜑, {ρ̂5, ρ̂7, ρ̂8, ρ̂9, ρ̂10, ρ̂11, ρ̂13}} {ρ̂4, ρ̂5, ρ̂6, ρ̂8, ρ̂9, ρ̂10, ρ̂12, ρ̂13} {𝜃(�̂�), 𝜑, {ρ̂4, ρ̂5, ρ̂6, ρ̂8, ρ̂9, ρ̂10, ρ̂12, ρ̂13}} 198 https://internationalpubls.com Table 4.5: Nano topologies�̂�𝑵𝒓 𝒊 [�̂�(�̂�)]by using initial right neighbourhoods forsubgraph of Figure 4.3 {ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂9, ρ̂10, ρ̂11, ρ̂12, ρ̂14} {𝜃(�̂�), 𝜑, {ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂9, ρ̂10, ρ̂11, ρ̂12, ρ̂14}} {ρ̂2, ρ̂3, ρ̂4, ρ̂5, ρ̂6, ρ̂7, ρ̂9, ρ̂11, ρ̂13} {𝜃(�̂�), 𝜑, {ρ̂2, ρ̂3, ρ̂4, ρ̂5, ρ̂6, ρ̂7, ρ̂9, ρ̂11, ρ̂13}} {ρ̂2, ρ̂3, ρ̂4, ρ̂5, ρ̂6, ρ̂8, ρ̂9, ρ̂11, ρ̂13, ρ̂14} {𝜃(�̂�), 𝜑, {ρ̂2, ρ̂3, ρ̂4, ρ̂5, ρ̂6, ρ̂8, ρ̂9, ρ̂11, ρ̂13, ρ̂14}} �̂�(�̂�) �̂�𝑵𝒓 𝒊 [�̂�(�̂�)] {ρ̂1,  ρ̂5} {𝜃(�̂�), 𝜑, {ρ̂1,  ρ̂5}} {ρ̂6,  ρ̂9} {𝜃(�̂�), 𝜑, {ρ̂6,  ρ̂9}} {ρ̂10,  ρ̂13} {𝜃(�̂�), 𝜑, {ρ̂10,  ρ̂13}} {ρ̂2,  ρ̂8} {𝜃(�̂�), 𝜑, {ρ̂2,  ρ̂8}} {ρ̂7,   ρ̂12} {𝜃(�̂�), 𝜑, {ρ̂7,   ρ̂12}} {ρ̂3, ρ̂6,   ρ̂11} {𝜃(�̂�), 𝜑, {ρ̂3, ρ̂6,   ρ̂11}} {ρ̂4, ρ̂6,   ρ̂12} {𝜃(�̂�), 𝜑, {ρ̂4, ρ̂6,   ρ̂12}} {ρ̂5, ρ̂7,   ρ̂13} {𝜃(�̂�), 𝜑, {ρ̂5, ρ̂7,   ρ̂13}} {ρ̂9, ρ̂12,  ρ̂14} {𝜃(�̂�), 𝜑, {ρ̂9, ρ̂12,  ρ̂14}} {ρ̂2, ρ̂3, ρ̂7, ρ̂8,   ρ̂9} {𝜃(�̂�), 𝜑, {ρ̂2, ρ̂3, ρ̂7, ρ̂8,   ρ̂9}} {ρ̂4, ρ̂8,  ρ̂9,   ρ̂11, ρ̂12} {𝜃(�̂�), 𝜑, {ρ̂4, ρ̂8,  ρ̂9,  ρ̂11, ρ̂12}} { ρ̂8,  ρ̂9, ρ̂10,  ρ̂11, ρ̂13} {𝜃(�̂�), 𝜑, { ρ̂8,  ρ̂9, ρ̂10,  ρ̂11, ρ̂13}} {ρ̂2, ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂9} {𝜃(�̂�), 𝜑, {ρ̂2, ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂9}} {ρ̂2, ρ̂4, ρ̂6, ρ̂7, ρ̂9, ρ̂10} {𝜃(�̂�), 𝜑, {ρ̂2, ρ̂4, ρ̂6, ρ̂7, ρ̂9, ρ̂10}} {ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂8, ρ̂9} {𝜃(�̂�), 𝜑, {ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂8, ρ̂9}} {ρ̂5, ρ̂6, ρ̂8,   ρ̂9,   ρ̂12, ρ̂14} {𝜃(�̂�), 𝜑, {ρ̂5, ρ̂6, ρ̂8,   ρ̂9,   ρ̂12, ρ̂14}} {ρ̂1, ρ̂2, ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂9} {𝜃(�̂�), 𝜑, {ρ̂1, ρ̂2, ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂9}} {ρ̂2, ρ̂3, ρ̂4, ρ̂6, ρ̂8, ρ̂9, ρ̂10} {𝜃(�̂�), 𝜑, {ρ̂2, ρ̂3, ρ̂4, ρ̂6, ρ̂8, ρ̂9, ρ̂10}} {ρ̂4, ρ̂5, ρ̂6, ρ̂8, ρ̂9, ρ̂10, ρ̂11} {𝜃(�̂�), 𝜑, {ρ̂4, ρ̂5, ρ̂6, ρ̂8, ρ̂9, ρ̂10, ρ̂11}} {ρ̂3, ρ̂5, ρ̂6, ρ̂9, ρ̂10, ρ̂11, ρ̂12} {𝜃(�̂�), 𝜑, {ρ̂3, ρ̂5, ρ̂6, ρ̂9, ρ̂10, ρ̂11, ρ̂12}} {ρ̂5, ρ̂7, ρ̂8, ρ̂9, ρ̂10, ρ̂11, ρ̂13} {𝜃(�̂�), 𝜑, {ρ̂5, ρ̂7, ρ̂8, ρ̂9, ρ̂10, ρ̂11, ρ̂13}} {ρ̂4, ρ̂5, ρ̂6, ρ̂8, ρ̂9, ρ̂10, ρ̂12, ρ̂13} {𝜃(�̂�), 𝜑, {ρ̂4, ρ̂5, ρ̂6, ρ̂8, ρ̂9, ρ̂10, ρ̂12, ρ̂13}} {ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂9, ρ̂10, ρ̂11, ρ̂12, ρ̂14} {𝜃(�̂�), 𝜑, {ρ̂3, ρ̂5, ρ̂6, ρ̂7, ρ̂9, ρ̂10, ρ̂11, ρ̂12, ρ̂14}} 199 https://internationalpubls.com 5.Discussion A theme of the current research work is that we have developed a digraph of the human double cycle system. From this we have derived certain nano-topological properties by creating initial left (resp. right) neighbourhoods and also we have identified the union and intersection of initial neighbourhoods might help us to ensure the cause of lung and heart failure. In addition we derived the nano-topologies by using initial left (resp. right) neighbourhoodsfor certain probable sub- graphs and also verified the comparison between these two types ofneighbourhoods.The findings of the digraph application approaches that we acquired are really helpful to ensure the functionality of double circulation system in the human body. Refrences [1] R. Abu-Gdairi, A. A. El-Atik, M. K. El-Bably . Topological visualization and graph analysis of rough sets via neighborhoods: A medical application using human heart data. AIMS Mathematics, 2023, 8(11): 26945-26967. doi: 10.3934/math.20231379 [2] Hakeem A. Othman, Mohammed M. 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