EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6694 ISSN 1307-5543 – ejpam.com Published by New York Business Global Topological Approaches of Graphs Using j-Neighbourhoods and Their Applications Amal T. Abushaaban1,∗, Abdelfattah A. El-Atik1, Osama A. Embaby1 1 Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt Abstract. This paper investigates novel topological structures on graphs through the lens of j- neighbourhoods, specifically out, in, intersection, and union-based neighbourhoods. We develop a systematic framework for constructing subbases and topologies on directed graphs using these neighbourhoods and analyze their topological properties. Our work provides a rigorous compara- tive study of neighbourhood types, their interrelations, and their role in generating induced topolo- gies. In addition, we explore potential applications in digital topology, spatial networks, and data structure. The theoretical results are supported by aircraft paths on an airline as an illustrative example and comparison tables that highlight structural differences and practical implications. 2020 Mathematics Subject Classifications: 05C10, 54A10, 05C20 Key Words and Phrases: Directed graphs, topological spaces, j-neighbourhoods, in-neighbourhood, out-neighbourhood, graph topology, subbase construction 1. Introduction The interplay between graph theory and topology has yielded powerful tools for mod- eling relationships in networks, decision systems, and digital structures. The foundation of topological graph theory, which seeks to apply topological concepts to graph structures, dates back to the foundational work of Kuratowski and others in the early twentieth century. This field has seen significant advances with the development of neighborhood systems and approximation spaces such as those initiated by Pawlak [1],[2] in rough set theory. Recent works have further expanded these concepts. Zhang et al. [3] and Lin [4] explored neighborhood operators in granular computing, while Yao [5] introduced rough sets based on covering with topological interpretations. Graph theory has recently estab- lished itself as an independent discipline [6, 7]. A graph GR = (V E,ED) is an ordered pair of vertices V E(GR) and edges ED(GR). We say that the graph GR is finite (resp. infinite) if the set V E(GR) is finite (resp. infinite). In medical decision-making systems, uncertain or imprecise concepts are often modeled using upper and lower approximations, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6694 Email addresses: amal_pg170074@science.tanta.edu.eg (A. Abushaaban), aelatik@science.tanta.edu.eg (A. El-Atik), embaby@science.tanta.edu.eg (O. Embaby) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 2 of 27 which serve as rough estimates of these vague entities[8, 9]. Akram et al. have con- tributed significantly to this field by exploring intuitionistic fuzzy and fuzzy rough graph structures[10–12], and applying them effectively to various decision-making scenarios. In [13], the solution of some problems in medicine and geography is deduced. A novel model for the blood circulation system of the human heart, structured around the charac- terization of blood flow pathways, is proposed by [14]. [15] Supplies a proof of the perfect graph theorem and features a revised chapter on the probabilistic method in graph theory with many results integrated throughout the text. Inspired by the diverse applications of general topology [16–27], our objective is to explore novel approaches to construct topo- logical structures through generalized neighborhood systems. Experiments show that the algorithm proposed in [28] is more efficient than the traditional one when the graph is represented by an adjacency matrix. Graph theory serves as a fundamental mathemati- cal framework that underpins a wide range of disciplines, including operational research, providing tools for modeling and solving complex relational structures [29, 30].Recent studies have extended topological approaches to graph structures, such as the work of Othman et al. [31] in L2-directed topological spaces for directed graphs, and Alzubaidi et al. [32] who investigated topologies on simple graphs with applications in radar chart methods. Our work builds on these foundations by developing a systematic framework using j-neighbourhoods to construct topological spaces on directed graphs. This paper begins with Section 2 which introduces the necessary preliminaries; Section 3 defines the j-neighborhood types; Section 4 discusses topological spaces on Air journies as an application; Section 5 includes conclusions and future work. 2. Preliminaries In this section, some definitions and propositions on graphs that will be used through- out this paper are stated, and their topologies are generated. Definition 1. [1] Let (V, S) be an approximation space, where V ̸= ϕ is a finite universe set, and S is an equivalence relation in V . Then: LOs(X) = ∪ x⊂V {S(x) : S(x) ⊆ X}, UPs(X) = ∩ x⊂V {S(x) : S(x) ∩ ̸= ϕ}. Definition 2. [13] Let (GR) be a graph with vertices V E(GR) and S be a relation to GR. The open neighbourhoods of (ve)i are (ve)iS = {(ve)j : −−−−−−→ (ve)i(ve)j ∈ E(GR)}. The subbase is SubGR = ∪ {(ve)iS : (ve)i ∈ V E(GR)} Definition 3. In [13], the topological space τGR in GR is induced by the base BGR that is induced by the subbase SubGR. Definition 4. [14] Let GR(V E,ED) be a simple directed graph, and ve ∈ V E(GR). The j-neighbourhoods of ve, say Nj(ve), j ∈ {t, n, int, un}, can be defined as: A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 3 of 27 (i) Out neighbourhood: Nt((ve)i) = ∪ i,k {(ve)i, (ve)k} ∪ {(ed)j : (ed)j = −−−−−−→ (ve)i(ve)k, (ed)j ∈ ED, (ve)i, (ve)k ∈ V E}, for each i, j, k ∈ I. (ii) In neighbourhood: Nn((ve)i) = ∪ i,k {(ve)i, (ve)k} ∪ {(ed)j : (ed)j = −−−−−−→ (ve)k(ve)i, (ed)j ∈ ED, (ve)i, (ve)k ∈ V E}, for each i, j, k ∈ I (iii) Intersection of neighbourhoods: Nint((ve)i) = Nt ∩ Nn. (iv) Union of neighbourhoods: Nun((ve)i) = Nt ∪ Nn. Proposition 1. [14] Let GR(V E,ED) be a simple digraph, Nj be different kinds of neighbourhoods, where j ∈ {t, n, int, un},K and M are two subgraphs of GR. Then: (i) LONj ((V E,ED)(K)) ⊆ (V E,ED)(K) ⊆ UPNj ((V E,ED)(K)); (ii) LONj (GR) = UPNj (GR) = GR; (iii) LONj (ϕ) = UPNj (ϕ) = ϕ; (iv) If (V E,ED)(K) ⊆ (V E,ED)(M), then LONj ((V E,ED)(K)) ⊆ LONj ((V E,ED)(M)) and UPNj ((V E,ED)(K)) ⊆ UPNj ((V E,ED)(M)); (v) (UPNj ((V E,ED)(K)))c = LONj ((V E,ED)(K))c, where ((V E,ED)(K))c is a com- plement to (V E,ED)(K); (vi) UPNj ((V E,ED)(K))c = (LONj ((V E,ED)(K)))c. Proposition 2. [14] Let GR(V E,ED) be a digraph, Nj(ve) be different kinds neighbour- hoods, where j ∈ {t, n, int, un},K and M be two subgraphs of GR. Then: (i) LONj ((V E,ED)(K)) ∪ LONj ((V E,ED)(M)) ⊆ LONj ((V E,ED)(K) ∪ (V E,ED)(M)); (ii) LONj ((V E,ED)(K) ∩ (V E,ED)(M)) = LONj ((V E,ED)(K)) ∩ LONj ((V E,ED)(M)); (iii) UPNj ((V E,ED)(K) ∪ (V E,ED)(M)) = UPNj ((V E,ED)(K)) ∪ UPNj ((V E,ED)(M)); (iv) UPNj ((V E,ED)(K) ∩ (V E,ED)(M)) ⊆ UPNj ((V E,ED)(K)) ∩ UPNj ((V E,ED)(M)). Proposition 3. [14] Let GR(V E,ED) be a simple digraph, Nj be different kinds of neighbourhoods, where j ∈ {t, n, int, un},K is a subgraph of GR. Then: (i) LONj ((V E,ED)(GR)− (V E,ED)(K)) = (V E,ED)(GR)− UPNj ((V E,ED)(K)); (ii) UPNj (V E,ED)(GR)− (V E,ED)(K)) = (V E,ED)(GR)− LONj ((V E,ED)(K)). Proposition 4. [14] Let GR(V E,ED) be a simple digraph Nj be different kinds of neigh- bourhoods, where j ∈ {t, n, int, un},K and M are two subgraphs of GR. Then: (i) LONj ((V E,ED)(K))−(V E,ED)(M)) ⊆ LONj ((V E,ED)(K))−LONj ((V E,ED)(M)); A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 4 of 27 (ii) UpNj ((V E,ED)(K))−UPNj ((V E,ED)(M)) ⊆ UPNj ((V E,ED)(K))−(V E,ED)(M)). Proposition 5. [14] Let GR(V E,ED) be a graph and ((V E,ED)(GR), τNj ) its nan- otopology K and M be induced subgraphs of GR. Then: (i) (V E,ED)(K) ⊆ SUNj ((V E,ED)(K)); (ii) If K ⊆ M , then SUNj ((V E,ED)(K)) ⊆ SUNj ((V E,ED)(M)); (iii) SUNj ((V E,ED)(K) ∪ (V E,ED)(M)) = SUNj ((V E,ED)(K)) ∪ SUNj ((V E,ED)(M)); (iv) SUNj ((V E,ED)(K) ∩ (V E,ED)(M)) ⊆ SUNj ((V E,ED)(K)) ∩ SUNj ((V E,ED)(M)); (v) If K ⊆ M then RINj ((V E,ED)(K)) ⊆ RINj ((V E,ED)(M)); (vi) RINj ((V E,ED)(K) ∪ RINj ((V E,ED)(M)) ⊆ RINj ((V E,ED)(K) ∪ (V E,ED)(M)); (vii) RINj ((V E,ED)(K) ∩ (V E,ED)(M)) = RINj ((V E,ED)(K) ∩ RINj ((V E,ED)(M)). Proposition 6. [14] Let GR(V E,ED) be a graph and ((V E,ED)(GR), τNj ) its nan- otopology, K be an induced subgraph of GR. Then: (i) RINj ((V E,ED)(GR)− (V E,ED)(K)) = (V E,ED)(GR)− SUNj ((V E,ED)(K)); (ii) SUNj ((V E,ED)(GR)− (V E,ED)(K)) = (V E,ED)(GR)−RINj ((V E,ED)(K)). 3. New Types of Topological Spaces by Simple Directed Graphs In this section, we introduce new kinds of j-neighbourhoods, j-lower(upper) approxi- mations with illustrative tables, in addition to how to find a subbase, a base, and generating topologies. Moreover, some propositions with related remarks will be exposed. The following definition defines and investigates four forms of neighbourhoods of vertices in simple directed graphs dependent on neighbouring vertices. Definition 5. Let SDG(L) be a simple directed graph, and l ∈ L(SDG). The j- neighbourhood of l, sayNj(l), j ∈ {t, n, int, un}, abbreviations for {out,in,intersection,union},can be defined as (i) Outside neighbourhood (briefly, t-neighbourhood): Nt(li) = {li ∈ L : i ̸= j, li ⇒ lj}, for each i, j ∈ I; (ii) Inside neighbourhood (briefly, n-neighbourhood): Nn(li) = {lj ∈ L : i ̸= j, lj ⇒ li}, for each i, j ∈ I; (iii) Intersection of neighbourhoods (briefly, int-neighbourhood): Nint(li) = Nt(li) ∩ Nn(li); (iv) Union of neighbourhoods (briefly, un-neighbourhood): Nun(li) = Nt(li) ∪ Nn(li). Example 1. Consider Figure 1. The set of vertices of a simple digraph is L(G) = {l1, l2, l3, l4}. A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 5 of 27 tableNeighbourhoods of simple digraph l ∈ L(SDG) l1 l2 l3 l4 Nt(li) {l3} {l1, l4} {l2} ϕ Nn(li) {l2} {l3} {l1} {l2} Nint(li) ϕ ϕ ϕ ϕ Nun(li) {l2, l3} {l1, l3, l4} {l1, l2} {l2} figureA simple directed graph Definition 6. Let SDG(L) be a simple directed graph, K be a subgraph of SDG and Nj(l), j ∈ {t, n, int, un} be the j-neighbourhood of l ∈ L(SDG). Then: (i) The j-lower approximation LONj ((L)(K)) = ∪ l∈(L)(SDG) {l : Nj(l) ⊆ (L)(K)}; (ii) The j-upper approximation UPNj ((L)(K)) = ∪ l∈(L)(SDG) {l : Nj(l) ∩ (L)(K) ̸= ϕ}. Example 2. From Example 1. We get the following tables. Table 1: LONj ((L)(K)) w.r.t. Table 1 L(K) LONt(L)(K) LONn(L)(K) LONint(L)(K) LONun(L)(K) ϕ {l4} ϕ L(SDG) ϕ L(SDG) L(SDG) L(SDG) L(SDG) L(SDG) {l1} {l4} {l3} L(SDG) ϕ {l2} {l3, l4} {l1, l4} L(SDG) {l4} {l3} {l1, l4} {l2} L(SDG) ϕ {l4} {l4} ϕ L(SDG) ϕ {l1, l2} {l3, l4} {l1, l3, l4} L(SDG) {l3, l4} {l1, l3} {l1, l4} {l2, l3} L(SDG) ϕ {l1, l4} {l2, l4} {l3} L(SDG) ϕ {l2, l3} {l1, l3, l4} {l1, l2, l4} L(SDG) {l1, l4} {l2, l4} {l3, l4} {l1, l4} L(SDG) {l4} {l3, l4} {l1, l4} {l2} L(SDG) ϕ {l1, l2, l3} {l1, l3, l4} L(SDG) L(SDG) {l1, l3, l4} {l1, l2, l4} {l2, l3, l4} {l1, l3, l4} L(SDG) {l3, l4} {l1, l3, l4} {l1, l2, l4} {l2, l3} L(SDG) {l2} {l2, l3, l4} {l1, l3, l4} {l1, l2, l4} L(SDG) {l1, l4} A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 6 of 27 Table 2: UPNj ((L)(K)) w.r.t. Table 1 L(K) UPNt(L)(K) UPNn(L)(K) UPNint(L)(K) UPNun(L)(K) ϕ ϕ ϕ ϕ ϕ L(SDG) {l1, l2, l3} L(SDG) ϕ L(SDG) {l1} {l2} {l3} ϕ {l2, l3} {l2} {l3} {l1, l4} ϕ {l1, l3, l4} {l3} {l1} {l2} ϕ {l1, l2} {l4} {l2} ϕ ϕ {l2} {l1, l2} {l2, l3} {l1, l3, l4} ϕ L(SDG) {l1, l3} {l1, l2} {l2, l3} ϕ {l1, l2, l3} {l1, l4} {l2} {l3} ϕ {l2, l3} {l2, l3} {l1, l3} {l1, l2, l4} ϕ L(SDG) {l2, l4} {l2, l3} {l1, l4} ϕ L(SDG) {l3, l4} {l1, l2} {l2} ϕ {l1, l2} {l1, l2, l3} {l1, l2, l3} L(SDG) ϕ L(SDG) {l1, l2, l4} {l2, l3} {l1, l3, l4} ϕ L(SDG) {l1, l3, l4} {l1, l2} {l2, l3} ϕ {l1, l2, l3} {l2, l3, l4} {l1, l2, l3} {l1, l2, l4} ϕ L(SDG) Definition 7. Let SDG(L) be a simple directed graph (L)(K) be a subgraph of SDG,Nj(l) be different kinds of j-neighbourhoods, where j ∈ {t, n, int, un} and Subj(L)(K) = {LONj (L)(K), UPNj (L)(K)} be a subbase where the base Bj(L)(K) be the finite intersection of the subbase elements and τNj ((L)(K)) be the topology on (L)(SDG) with respect to (L)(K) which can be find by an arbitrary unions of the base elements ((L)(SDG), τNj (L)(K)) is a topology induced by different j-neighbourhoods. The topology τ = ∩ {τNj , j ∈ {t, n, int, un}}. Example 3. From Example 2, we get the subbase Subj(L)(K) in Table 4, the base Bj(L)(K) in Table 5 and τNj (L)(K) in Table 6. * Note that τ = {ϕ,L(SDG)} for any (L)(K). A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 7 of 27 Table 3: Subj(L)(K) L(K) Subt(L)(K) Subn(L)(K) Subint(L)(K) Subun(L)(K) ϕ {ϕ, {l4}} {ϕ} {ϕ,L(SDG)} {ϕ} L(SDG) {L(SDG), {l1, l2, l3}} {L(SDG)} {ϕ,L(SDG)} {L(SDG)} {l1} {{l2}, {l4}} {{l3}} {ϕ,L(SDG)} {ϕ, {l2, l3}} {l2} {{l3}, {l3, l4}} {{l1, l4}} {ϕ,L(SDG)} {{l4}, {l1, l3, l4}} {l3} {{l1}, {l1, l4}} {{l2}} {ϕ,L(SDG)} {ϕ, {l1, l2}} {l4} {{l2}, {l4}} {ϕ} {ϕ,L(SDG)} {ϕ, {l2}} {l1, l2} {{l2, l3}, {l3, l4}} {{l1, l3, l4}} {ϕ,L(SDG)} {{l3, l4}, L(SDG)} {l1, l3} {{l1, l2}, {l1, l4}} {{l2, l3}} {ϕ,L(SDG)} {ϕ, {l1, l2, l3}} {l1, l4} {{l2}, {l2, l4}} {{l3}} {ϕ,L(SDG)} {ϕ, {l2, l3}} {l2, l3} {{l1, l3}, {l1, l3, l4}} {{l1, l2, l4}} {ϕ,L(SDG)} {{l1, l4}, L(SDG)} {l2, l4} {{l2, l3}, {l3, l4}} {{l1, l4}} {ϕ,L(SDG)} {{l4}, L(SDG)} {l3, l4} {{l1, l2}, {l1, l4}} {{l2}} {ϕ,L(SDG)} {ϕ, {l1, l2}} {l1, l2, l3} {{l1, l2, l3}, {l1, l3, l4}} {L(SDG)} {ϕ,L(SDG)} {L(SDG), {l1, l3, l4}} {l1, l2, l4} {{l2, l3}, {l2, l3, l4}} {{l1, l3, l4}} {ϕ,L(SDG)} {L(SDG), {l3, l4}} {l1, l3, l4} {{l1, l2}, {l1, l2, l4}} {{l2, l3}} {ϕ,L(SDG)} {{l2}, {l1, l2, l3}} {l2, l3, l4} {{l1, l2, l3}, {l1, l3, l4}} {{l1, l2, l4}} {ϕ,L(SDG)} {{l1, l4}, L(SDG)} Table 4: Bj(L)(K) L(K) Bt(L)(K) Bn(L)(K) Bint(L)(K) Bun(L)(K) ϕ {ϕ,L(SDG), {l4}} {L(SDG), ϕ} {ϕ,L(SDG)} {ϕ,L(SDG)} L(SDG) {L(SDG), {l1, l2, l3}} {L(SDG)} {ϕ,L(SDG)} {L(SDG)} {l1} {ϕ,L(SDG), {l2}, {l4}} {{L(SDG), {l3}} {ϕ,L(SDG)} {ϕ,L(SDG), {l2, l3}} {l2} {L(SDG), {l3}, {l3, l4}} {L(SDG), {l1, l4}} {ϕ,L(SDG)} {L(SDG), {l4}, {l1, l3, l4}} {l3} {L(SDG), {l1}, {l1, l4}} {L(SDG), {l2}} {ϕ,L(SDG)} {L(SDG), ϕ, {l1, l2}} {l4} {L(SDG), ϕ, {l2}, {l4}} {ϕ,L(SDG)} {ϕ,L(SDG)} {L(SDG), ϕ, {l2}} {l1, l2} {L(SDG), {l3}, {l2, l3}, {l3, l4}} {L(SDG), {l1, l3, l4}} {ϕ,L(SDG)} {{l3, l4}, L(SDG)} {l1, l3} {L(SDG), {l1}, {l1, l2}, {l1, l4}} {L(SDG), {l2, l3}} {ϕ,L(SDG)} {L(SDG), ϕ, {l1, l2, l3}} {l1, l4} {L(SDG), {l2}, {l2, l4}} {L(SDG), {l3}} {ϕ,L(SDG)} {L(SDG), ϕ, {l2, l3}} {l2, l3} {L(SDG), {l1, l3}, {l1, l3, l4}} {L(SDG), {l1, l2, l4}} {ϕ,L(SDG)} {{l1, l4}, L(SDG)} {l2, l4} {L(SDG), {l3}{l2, l3}, {l3, l4}} {L(SDG), {l1, l4}} {ϕ,L(SDG)} {{l4}, L(SDG)} {l3, l4} {L(SDG), {l1}, {l1, l2}, {l1, l4}} {L(SDG), {l2}} {ϕ,L(SDG)} {L(SDG), ϕ, {l1, l2}} {l1, l2, l3} {L(SDG), {l1, l3}, {l1, l2, l3}, {l1, l3, l4}} {L(SDG)} {ϕ,L(SDG)} {L(SDG), {l1, l3, l4}} {l1, l2, l4} {L(SDG), {l2, l3}, {l2, l3, l4}} {L(SDG), {l1, l3, l4}} {ϕ,L(SDG)} {L(SDG), {l3, l4}} {l1, l3, l4} {L(SDG), {l1, l2}, {l1, l2, l4}} {L(SDG), {l2, l3}} {ϕ,L(SDG)} {L(SDG), {l2}, {l1, l2, l3}} {l2, l3, l4} {L(SDG), {l1, l3}, {l1, l2, l3}, {l1, l3, l4}} {L(SDG), {l1, l2, l4}} {ϕ,L(SDG)} {{l1, l4}, L(SDG)} A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 8 of 27 Table 5: τj(L)(K) L(K) τNt(L)(K) τNn(L)(K) τNint(L)(K) τNun(L)(K) ϕ {ϕ,L(SDG), {l4}} {L(SDG), ϕ} {ϕ,L(SDG)} {ϕ,L(SDG)} L(SDG) {ϕ,L(SDG), {l1, l2, l3}} {ϕ,L(SDG)} {ϕ,L(SDG)} {ϕ,L(SDG)} {l1} {ϕ,L(SDG), {l2}, {l4}, {l2, l4}} {ϕ, {L(SDG), {l3}} {ϕ,L(SDG)} {ϕ,L(SDG), {l2, l3}} {l2} {ϕ,L(SDG), {l3}, {l3, l4}} {ϕ,L(SDG), {l1, l4}} {ϕ,L(SDG)} {ϕ,L(SDG), {l4}, {l1, l3, l4}} {l3} {ϕ,L(SDG), {l1}, {l1, l4}} {ϕ,L(SDG), {l2}} {ϕ,L(SDG)} {L(SDG), ϕ, {l1, l2}} {l4} {L(SDG), ϕ, {l2}, {l4}, {l2, l4}} {ϕ,L(SDG)} {ϕ,L(SDG)} {L(SDG), ϕ, {l2}} {l1, l2} {ϕ,L(SDG), {l3}, {l2, l3}, {l3, l4}, {l2, l3, l4}} {ϕ,L(SDG), {l1, l3, l4}} {ϕ,L(SDG)} {ϕ, {l3, l4}, L(SDG)} {l1, l3} {ϕ,L(SDG), {l1}, {l1, l2}, {l1, l4}, {l1, l2, l4}} {ϕ,L(SDG), {l2, l3}} {ϕ,L(SDG)} {L(SDG), ϕ, {l1, l2, l3}} {l1, l4} {ϕ,L(SDG), {l2}, {l2, l4}} {ϕ,L(SDG), {l3}} {ϕ,L(SDG)} {L(SDG), ϕ, {l2, l3}} {l2, l3} {ϕ,L(SDG), {l1, l3}, {l1, l3, l4}} {ϕ,L(SDG), {l1, l2, l4}} {ϕ,L(SDG)} {ϕ, {l1, l4}, L(SDG)} {l2, l4} {ϕ,L(SDG), {l3}{l2, l3}, {l3, l4}, {l2, l3, l4}} {ϕ,L(SDG), {l1, l4}} {ϕ,L(SDG)} {ϕ, {l4}, L(SDG)} {l3, l4} {ϕ,L(SDG), {l1}, {l1, l2}, {l1, l4}, {l1, l2, l4}} {ϕ,L(SDG), {l2}} {ϕ,L(SDG)} {L(SDG), ϕ, {l1, l2}} {l1, l2, l3} {ϕ,L(SDG), {l1, l3}, {l1, l2, l3}, {l1, l3, l4}} {ϕ,L(SDG)} {ϕ,L(SDG)} {ϕ,L(SDG), {l1, l3, l4}} {l1, l2, l4} {ϕ,L(SDG), {l2, l3}, {l2, l3, l4}} {ϕ,L(SDG), {l1, l3, l4}} {ϕ,L(SDG)} {ϕ,L(SDG), {l3, l4}} {l1, l3, l4} {ϕ,L(SDG), {l1, l2}, {l1, l2, l4}} {ϕ,L(SDG), {l2, l3}} {ϕ,L(SDG)} {ϕ,L(SDG), {l2}, {l1, l2, l3}} {l2, l3, l4} {ϕ,L(SDG), {l1, l3}, {l1, l2, l3}, {l1, l3, l4}} {ϕ,L(SDG), {l1, l2, l4}} {ϕ,L(SDG)} {ϕ, {l1, l4}, L(SDG)} Table 6: (LONj ((L)(K)))c w.r.t. Table 2 L(K) (LONt(L)(K))c (LONn(L)(K))c (LONint(L)(K))c (LONun(L)(K))c ϕ {l1, l2, l3} L(SDG) ϕ L(SDG) L(SDG) ϕ ϕ ϕ ϕ {l1} {l1, l2, l3} {l1, l2, l4} ϕ L(SDG) {l2} {l1, l2} {l2, l3} ϕ {l1, l2, l3} {l3} {l2, l3} {l1, l3, l4} ϕ L(SDG) {l4} {l1, l2, l3} L(SDG) ϕ L(SDG) {l1, l2} {l1, l2} {l2} ϕ {l1, l2} {l1, l3} {l2, l3} {l1, l4} ϕ L(SDG) {l1, l4} {l1, l3} {l1, l2, l4} ϕ L(SDG) {l2, l3} {l2} {l3} ϕ {l2, l3} {l2, l4} {l1, l2} {l2, l3} ϕ {l1, l2, l3} {l3, l4} {l2, l3} {l1, l3, l4} ϕ L(SDG) {l1, l2, l3} {l2} ϕ ϕ {l2} {l1, l2, l4} {l1} {l2} ϕ {l1, l2} {l1, l3, l4} {l3} {l1, l4} ϕ {l1, l3, l4} {l2, l3, l4} {l2} {l3} ϕ {l2, l3} A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 9 of 27 Table 7: (UPNj ((L)(K)))c w.r.t. Table 3 L(K) (UPNt(L)(K))c (UPNn(L)(K))c (UPNint(L)(K))c (UPNun(L)(K))c ϕ L(SDG) L(SDG) L(SDG) L(SDG) L(SDG) {l4} ϕ L(SDG) ϕ {l1} {l1, l3, l4} {l1, l2, l4} L(SDG) {l1, l4} {l2} {l1, l2, l4} {l2, l3} L(SDG) {l2} {l3} {l2, l3, l4} {l1, l3, l4} L(SDG) {l3, l4} {l4} {l1, l3, l4} L(SDG) L(SDG) {l1, l3, l4} {l1, l2} {l1, l4} {l2} L(SDG) ϕ {l1, l3} {l3, l4} {l1, l4} L(SDG) {l4} {l1, l4} {l1, l3, l4} {l1, l2, l4} L(SDG) {l1, l4} {l2, l3} {l2, l4} {l3} L(SDG) ϕ {l2, l4} {l1, l4} {l2, l3} L(SDG) ϕ {l3, l4} {l3, l4} {l1, l3, l4} L(SDG) {l3, l4} {l1, l2, l3} {l4} ϕ L(SDG) ϕ {l1, l2, l4} {l1, l4} {l2} L(SDG) ϕ {l1, l3, l4} {l3, l4} {l1, l4} L(SDG) {l4} {l2, l3, l4} {l4} {l3} L(SDG) ϕ Table 8: LONj ((L)(K))c (L(K))c LONt((L)(K))c LONn((L)(K))c LONint((L)(K))c LONun((L)(K))c L(SDG) L(SDG) L(SDG) L(SDG) L(SDG) ϕ {l4} ϕ L(SDG) ϕ {l2, l3, l4} {l1, l3, l4} {l1, l2, l4} L(SDG) {l1, l4} {l1, l3, l4} {l1, l2, l4} {l2, l3} L(SDG) {l2} {l1, l2, l4} {l2, l3, l4} {l1, l3, l4} L(SDG) {l3, l4} {l1, l2, l3} {l1, l3, l4} L(SDG) L(SDG) {l1, l3, l4} {l3, l4} {l1, l4} {l2} L(SDG) ϕ {l2, l4} {l3, l4} {l1, l4} L(SDG) {l4} {l2, l3} {l1, l3, l4} {l1, l2, l4} L(SDG) {l1, l4} {l1, l4} {l2, l4} {l3} L(SDG) ϕ {l1, l3} {l1, l4} {l2, l3} L(SDG) ϕ {l1, l2} {l3, l4} {l1, l3, l4} L(SDG) {l3, l4} {l4} {l4} ϕ L(SDG) ϕ {l3} {l1, l4} {l2} L(SDG) ϕ {l2} {l3, l4} {l1, l4} L(SDG) {l4} {l1} {l4} {l3} L(SDG) ϕ A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 10 of 27 Table 9: UPNj ((L)(K))c (L(K))c UPNt((L)(K))c UPNn((L)(K))c UPNint((L)(K))c UPNun((L)(K))c L(SDG) {l1, l2, l3} L(SDG) ϕ L(SDG) ϕ ϕ ϕ ϕ ϕ {l2, l3, l4} {l1, l2, l3} {l1, l2, l4} ϕ L(SDG) {l1, l3, l4} {l1, l2} {l2, l3} ϕ {l1, l2, l3} {l1, l2, l4} {l2, l3} {l1, l3, l4} ϕ L(SDG) {l1, l2, l3} {l1, l2, l3} L(SDG) ϕ L(SDG) {l3, l4} {l1, l2} {l2} ϕ {l1, l2} {l2, l4} {l2, l3} {l1, l4} ϕ L(SDG) {l2, l3} {l1, l3} {l1, l2, l4} ϕ L(SDG) {l1, l4} {l2} {l3} ϕ {l2, l3} {l1, l3} {l1, l2} {l2, l3} ϕ {l1, l2, l3} {l1, l2} {l2, l3} {l1, l3, l4} ϕ L(SDG) {l4} {l2} ϕ ϕ {l2} {l3} {l1} {l2} ϕ {l1, l2} {l2}} {l3} {l1, l4} ϕ {l1, l3, l4} {l1} {l2} {l3} ϕ {l2, l3} Definition 8. Let SDG(L) be a simple directed graph, and let (L)(K) be a subgraph of SDG. Then the j-closure(j-interior) of K is the j-upper(j-lower) approximation of K and can be defined by SUNj (L)(K) = UPNj (L)(K)(RINj (L)(K) = LONj (L)(K)), where j ∈ {t, n, int, un}. Proposition 7. Let SDG(L) be a simple directed graph, Nj be different kinds of j-neighbourhoods, where j ∈ {t, n, int, un},K and M are two subgraphs of SDG. Then: (i) LONj (SDG) = SDG; (ii) UPNj (ϕ) = ϕ; (iii) If (L)(K) ⊆ (L)(M), then LONj (L)(K) ⊆ LONj (L)(M) and UPNj (L)(K) ⊆ UPNj (L)(M); (iv) (UPNj (L)(K))c = LONj ((L)(K))c, where ((L)(K))c is a complement of (L)(K); (v) UPNj ((L)(K))c = (LONj ((l)(K)))c. Proof. (i) and (ii) are obvious from Definition 6. (iii) Let l ∈ LONj ((L)(K)). Then Nj(l) ⊆ (L)(K) by Definition 6, but (L)(K) ⊆ (L)(M). Then Nj(l) ⊆ (L)(M) and so l ∈ LONj (L)(M). Therefore, LONj (L)(K) ⊆ LONj (L)(M). The proof of UPNj (L)(K) ⊆ UPNj (L)(M) is similar. (iv) (UPNj (L)(K))c = ( ∪ l∈(L)(SDG) {l : Nj(l) ∩ (L)(K) ̸= ϕ})c = {l ∈ (L)(SDG) : Nj(l) ∩ (L)(K) = ϕ} = {l ∈ (L)(SDG) : Nj(l) ⊆ ((L)(K))c} = LONj ((L)(K))c. (v) Similar to (iv). Remark 1. Let SDG(L) be a simple directed graph. Nj are different kinds of j- neighbourhoods, where j ∈ {t, n, int, un},K is a subgraph of SDG. Then the following: (i) LONj (L)(K) ⊆ (L)(K) ⊆ UPNj (L)(K); A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 11 of 27 (ii) UPNj (SDG) = SDG,LONj (ϕ) = ϕ; are not always true. Example 4. According to Tables 2 and 3, (i) LONt({l1}) = {l4} ⊈ {l1} ⊈ {l2} = UPNt({l1}); (ii) UPNt(SDG) = {l1, l2, l3} ̸= SDG,LONt(ϕ) = {l4} ̸= ϕ. Proposition 8. Let SDG(L) be a simple directed graph, Nj(l) be different kinds of neigh- bourhoods, where j ∈ {t, n, int, un},K and M be two subgraphs of SDG. Then: (i) LONj (L)(K) ∪ LONj (L)(M) ⊆ LONj ((L)(K) ∪ (L)(M)); (ii) LONj ((L)(K) ∩ (L)(M)) = LONj (L)(K) ∩ LONj (L)(M); (iii) UPNj ((L)(K) ∪ (L)(M)) = UPNj (L)(K) ∪ UPNj (L)(M); (iv) UPNj ((L)(K) ∩ (L)(M)) ⊆ UPNj (L)(K) ∩ UPNj (L)(M). Proof. (i) As (L)(K) ⊆ (L)(K) ∪ (L)(M) and (L)(M) ⊆ (L)(K) ∪ (L)(M), then LONj (L)(K) ⊆ LONj ((L)(K) ∪ (L)(M)) and LONj (L)(M) ⊆ LONj ((L)(K) ∪ (L)(M)). Hence, LONj (L)(K) ∪ LONj (L)(M) ⊆ LONj ((L)(K) ∪ (L)(M)). (ii) As (L)(K) ∩ (L)(M) ⊆ (L)(K) and (L)(K) ∩ (L)(M) ⊆ (L)(M), then LONj ((L)(K) ∩ (L)(M)) ⊆ LONj (L)(K) and LONj ((L)(K) ∩ (L)(M)) ⊆ LONj (L)(M). So, LONj ((L)(K) ∩ (L)(M)) ⊆ LONj (L)(K) ∩ LONj (L)(M). Let l ∈ LONj (L)(K) ∩ LONj (L)(M), then l ∈ LONj (L)(K) and l ∈ LONj (L)(M). By Definition 6(i), Nj(l) ⊆ (L)(K) and Nj(l) ⊆ (L)(M), and thus Nj(l) ⊆ (L)(K) ∩ (L)(M). So, l ∈ LONj ((L)(K)∩ (L)(M)). Therefore, LONj (L)(K) ∩ LONj (L)(M) ⊆ LONj ((L)(K) ∩ (L)(M)). Hence, the result. (iii) As (L)(K) ⊆ (L)(K) ∪ (L)(M) and (L)(M) ⊆ (L)(K) ∪ (L)(M), then UPNj (L)(K) ⊆ UPNj ((L)(K) ∪ (L)(M)) and UPNj (L)(M) ⊆ UPNj ((L)(K) ∪ (L)(M)). So, UPNj (L)(K)∪ UPNj (L)(M) ⊆ UPNj ((L)(K) ∪ (L)(M)). Let l ∈ UPNj ((L)(K) ∪ (L)(M)). Then by Definition 6 (ii), l ∈ ∪ l∈L(SDG) {l : Nj(l) ∩ ((L)(K) ∪ (L)(M)) ̸= ϕ}, then l ∈∪ l∈L(SDG) {l : Nj(l) ∩ (L)(K) ̸= ϕ} or l ∈ ∪ l∈L(SDG) {l : Nj(l) ∩ (L)(M)) ̸= ϕ}, that is l ∈ UPNj (L)(K) or l ∈ UPNj (L)(M) so l ∈ UPNj (L)(K) ∪ UPNj (L)(M). Therefore, UPNj ((L)(K) ∪ (L)(M)) ⊆ UPNj (L)(K) ∪ UPNj (L)(M). Hence the result. (iv) As (L)(K) ∩ (L)(M) ⊆ (L)(K) and (L)(K) ∩ (L)(M) ⊆ (L)(M), then UPNj ((L)(K) ∩ (L)(M)) ⊆ UPNj (L)(K) and UPNj ((L)(K) ∩ (L)(M)) ⊆ UPNj (L)(M). Hence, the result. A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 12 of 27 Proposition 9. Let (SDG)(L) be a simple directed graph, Nj be different kinds of neigh- bourhoods where j ∈ {t, n, int, un},K be a subgraph of SDG. Then: (i) LONj ((L)(SDG)− (L)(K)) = (L)(SDG)− UPNj (L)(K); (ii) UPNj ((L)(SDG)− (L)(K)) = (L)(SDG)− LONj (L)(K). Proof. (i) As LONj ((L)(SDG)− (L)(K)) = LONj ((L)(K))c = (UPNj (L)(K))c = (L)(SDG)− UPNj (L)(K), by using Proposition 7(iv). (ii) As UPNj ((L)(SDG)− (L)(K)) = UPNj ((L)(K))c = (LONj (L)(K))c = (L)(SDG)− LONj (L)(K), using Proposition 7(v). Proposition 10. Let SDG(L) be a simple directed graph, Nj be different kinds of neighbourhoods where j ∈ {t, n, int, un},K and M are two subgraphs of SDG. Then: UPNj (L)(K)− UPj(L)(M) ⊆ UPNj ((L)(K)− (L)(M)). Proof. Let l ∈ (UPNj (L)(K) − UPNj (L)(M)), then by Definition 6, l ∈ ∪ {l : Nj(l) ∩ (L)(K) ̸= ϕ} and l /∈ ∪ {l : Nj(l) ∩ (L)(M) ̸= ϕ}. That is l ∈ ∪ {l : Nj(l) ∩ ((L)(M))c ̸= ϕ}. Therefore, l ∈ ∪ {l : Nj(l) ∩ [(L)(K) ∩ ((L)(M))c] ̸= ϕ}. So, l ∈ ∪ {l : Nj(l) ∩ [(L)(K) − (L)(M)] ̸= ϕ}. Hence, l ∈ UPNj ((L)(K)− (L)(M)) Then the result. Remark 2. Let SDG(L) be a simple directed graph, Nj be different kinds of neighbourhoods where j ∈ {t, n, int, un},K and M are two subgraphs of SDG. Then LONj ((L)(K) − (L)(M)) ⊈ LONj (L)(K)− LONj (L)(M). Example 5. From Table 2, LONt({l1, l2} − {l1, l3}) = LONt({l2}) = {l3, l4} ⊈ {l3} = {l3, l4} − {l1, l4} = LONt({l1, l2})− LONt(l1, l3}). From Definition 8 and Propositions 7, 8 and 9,we obtain the following. Proposition 11. Let SDG(L) be a simple directed graph, (L)(K) and (L)(M) be subgraphs of SDG. Then: (i) If (L)(K) ⊆ (L)(M), then SUNj (L)(K) ⊆ SUNj (L)(M) and RINj (L)(K) ⊆ RINj (L)(M); (ii) SUNj ((L)(K) ∪ (L)(M)) = SUNj (L)(K) ∪ SUNj (L)(M); (iii) SUNj ((L)(K) ∩ (L)(M)) ⊆ SUNj (L)(K) ∩ SUNj (L)(M); (iv) RINj (L)(K) ∪ RINj (L)(M) ⊆ RINj ((L)(K) ∪ (L)(M)); (v) RINj ((L)(K) ∩ (L)(M)) = RINj (L)(K) ∩ RINj (L)(M); (vi) RINj ((L)(SDG)− (L)(K)) = (L)(SDG)− SUNj (L)(K); (vii) SUNj ((L)(SDG)− (L)) = (L)(SDG)−RINj (L)(K). A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 13 of 27 4. An Application to Air Traffic Networks In this section, depending on new kinds of j-neighbourhoods and their topological spaces generated by them, we present various air journies ranking via these. The illustra- tion illustrates one of the air line companies’ routing plan for aircraft. Figure 1: A simple digraph of air journies Table 10: Neighbourhoods of simple digraph of the air journeys l ∈ L(SDG) l1 l2 l3 l4 l5 Nt(li) {l2, l3} {l5} {l4} {l2} ϕ Nn(li) ϕ {l1, l4} {l1} {l3} {l2} Nint(li) ϕ ϕ ϕ ϕ ϕ Nun(li) {l2, l3} {l1, l4, l5} {l1, l4} {l2, l3} {l2} Table 11: LONj ((L)(K)) with respect to Table 11 L(K) LONt(L)(K) LONn(L)(K) LONint(L)(K) LONun(L)(K) ϕ {l5} {l1} L(SDG) ϕ L(SDG) L(SDG) L(SDG) L(SDG) L(SDG) {l1} {l5} {l1, l3} L(SDG) ϕ {l2} {l4, l5} {l1, l5} L(SDG) {l5} {l3} {l5} {l1, l4} L(SDG) ϕ {l4} {l3, l5} {l1} L(SDG) ϕ {l5} {l2, l5} {l1} L(SDG) ϕ {l1, l2} {l4, l5} {l1, l3, l5} L(SDG) {l5} {l1, l3} {l5} {l1, l3, l4} L(SDG) ϕ Continued on next page A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 14 of 27 Table 11 – Continued from previous page L(K) LONt(L)(K) LONn(L)(K) LONint(L)(K) LONun(L)(K) {l1, l4} {l3, l5} {l1, l2, l3} L(SDG) {l3} {l1, l5} {l2, l5} {l1, l3} L(SDG) ϕ {l2, l3} {l1, l4, l5} {l1, l4, l5} L(SDG) {l1, l4, l5} {l2, l4} {l3, l4, l5} {l1, l5} L(SDG) {l5} {l2, l5} {l2, l4, l5} {l1, l5} L(SDG) {l5} {l3, l4} {l3, l5} {l1, l4} L(SDG) ϕ {l3, l5} {l2, l5} {l1, l4} L(SDG) ϕ {l4, l5} {l2, l3, l5} {l1} L(SDG) ϕ {l1, l2, l3} {l1, l4, l5} {l1, l3, l4, l5} L(SDG) {l1, l4, l5} {l1, l2, l4} {l3, l4, l5} {l1, l2, l3, l5} L(SDG) {l3, l5} {l1, l2, l5} {l2, l4, l5} {l1, l3, l5} L(SDG) {l5} {l1, l3, l4} {l3, l5} {l1, l2, l3, l4} L(SDG) {l3} {l1, l3, l5} {l2, l5} {l1, l3, l4} L(SDG) ϕ {l1, l4, l5} {l2, l3, l5} {l1, l2, l3} L(SDG) {l2, l3} {l2, l3, l4} {l1, l3, l4, l5} {l1, l4, l5} L(SDG) {l1, l4, l5} {l2, l3, l5} {l1, l2, l4, l5} {l1, l4, l5} L(SDG) {l1, l4, l5} {l2, l4, l5} {l2, l3, l4, l5} {l1, l5} L(SDG) {l5} {l3, l4, l5} {l2, l3, l5} {l1, l4} L(SDG) ϕ {l1, l2, l3, l4} {l1, l3, l4, l5} L(SDG) L(SDG) {l1, l3, l4, l5} {l1, l2, l3, l5} {l1, l2, l4, l5} {l1, l3, l4, l5} L(SDG) {l1, l4, l5} {l1, l2, l4, l5} {l2, l3, l4, l5} {l1, l2, l3, l5} L(SDG) {l2, l3, l5} {l1, l3, l4, l5} {l2, l3, l5} {l1, l2, l3, l4} L(SDG) {l3} {l2, l3, l4, l5} L(SDG) {l1, l4, l5} L(SDG) {l1, l4, l5} Table 12: UPNj ((L)(K)) with respect to Table 11 L(K) UPNt(L)(K) UPNn(L)(K) UPNint(L)(K) UPNun(L)(K) ϕ ϕ ϕ ϕ ϕ L(SDG) {l1, l2, l3, l4} {l2, l3, l4, l5} ϕ L(SDG) {l1} ϕ {l2, l3} ϕ {l2, l3} {l2} {l1, l4} {l5} ϕ {l1, l4, l5} {l3} {l1} {l4} ϕ {l1, l4} {l4} {l3} {l2} ϕ {l2, l3} {l5} {l2} ϕ ϕ {l2} {l1, l2} {l1, l4} {l2, l3, l5} ϕ L(SDG) {l1, l3} {l1} {l2, l3, l4} ϕ {l1, l2, l3, l4} {l1, l4} {l3} {l2, l3} ϕ {l2, l3} {l1, l5} {l2} {l2, l3} ϕ {l2, l3} {l2, l3} {l1, l4} {l4, l5} ϕ {l1, l4, l5} Continued on next page A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 15 of 27 Table 12 – Continued from previous page L(K) UPNt(L)(K) UPNn(L)(K) UPNint(L)(K) UPNun(L)(K) {l2, l4} {l1, l3, l4} {l2, l5} ϕ L(SDG) {l2, l5} {l1, l2, l4} {l5} ϕ {l1, l2, l4, l5} {l3, l4} {l1, l3} {l2, l4} ϕ {l1, l2, l3, l4} {l3, l5} {l1, l2} {l4} ϕ {l1, l2, l4} {l4, l5} {l2, l3} {l2} ϕ {l2, l3} {l1, l2, l3} {l1, l4} {l2, l3, l4, l5} ϕ L(SDG) {l1, l2, l4} {l1, l3, l4} {l2, l3, l5} ϕ L(SDG) {l1, l2, l5} {l1, l2, l4} {l2, l3, l5} ϕ L(SDG) {l1, l3, l4} {l1, l3} {l2, l3, l4} ϕ {l1, l2, l3, l4} {l1, l3, l5} {l1, l5} {l2, l3, l4} ϕ {l1, l2, l3, l4} {l1, l4, l5} {l2, l3} {l2, l3} ϕ {l2, l3} {l2, l3, l4} {l1, l3, l4} {l2, l4, l5} ϕ L(SDG) {l2, l3, l5} {l1, l2, l4} {l4, l5} ϕ {l1, l2, l4, l5} {l2, l4, l5} {l1, l2, l3, l4} {l2, l5} ϕ L(SDG) {l3, l4, l5} {l1, l2, l3} {l2, l4} ϕ {l1, l2, l3, l4} {l1, l2, l3, l4} {l1, l3, l4} {l2, l3, l4, l5} ϕ L(SDG) {l1, l2, l3, l5} {l1, l2, l4} {l2, l3, l4, l5} ϕ L(SDG) {l1, l2, l4, l5} {l1, l2, l3, l4} {l2, l3, l5} ϕ L(SDG) {l1, l3, l4, l5} {l1, l2, l3} {l2, l3, l4} ϕ {l1, l2, l3, l4} {l2, l3, l4, l5} {l1, l2, l3, l4} {l2, l4, l5} ϕ L(SDG) Table 13: Subt(L)(K) and Subn(L)(K) with respect to Ta- bles 12 and 13 L(K) Subt(L)(K) Subn(L)(K) ϕ {ϕ, {l5}} {ϕ, {l1}} L(SDG) {{l1, l2, l3, l4}, L(SDG)} {{l2, l3, l4, l5}, L(SDG)} {l1} {ϕ, {l5}} {{l1, l3}, {l2, l3}} {l2} {{l1, l4}, {l4, l5}} {{l5}, {l1, l5}} {l3} {{l1}, {l5}} {{l4}, {l1, l4}} {l4} {{l3}, {l3, l5}} {{l1}, {l2}} {l5} {{l2}, {l2, l5}} {ϕ, {l1}} {l1, l2} {{l1, l4}, {l4, l5}} {{l1, l3, l5}, {l2, l3, l5}} {l1, l3} {{l1}, {l5}} {{l1, l3, l4}, {l2, l3, l4}} {l1, l4} {{l3}, {l3, l5}} {{l2, l3}, {l1, l2, l3}} {l1, l5} {{l2}, {l2, l5}} {{l1, l3}, {l2, l3}} {l2, l3} {{l1, l4}, {l1, l4, l5}} {{l4, l5}, {l1, l4, l5}} {l2, l4} {{l1, l3, l4}, {l3, l4, l5}} {{l2, l5}, {l1, l5}} {l2, l5} {{l1, l2, l4}, {l2, l4, l5}} {{l5}, {l1, l5}} Continued on next page A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 16 of 27 Table 13 – Continued from previous page L(K) Subt(L)(K) Subn(L)(K) {l3, l4} {{l1, l3}, {l3, l5}} {{l1, l4}, {l2, l4}} {l3, l5} {{l1, l2}, {l2, l5}} {{l4}, {l1, l4}} {l4, l5} {{l2, l3}, {l2, l3, l5}} {{l1}, {l2}} {l1, l2, l3} {{l1, l4}, {l1, l4, l5}} {{l1, l3, l4, l5}, {l2, l3, l4, l5}} {l1, l2, l4} {{l1, l3, l4}, {l3, l4, l5}} {{l1, l2, l3, l5}, {l2, l3, l5}} {l1, l2, l5} {{l1, l2, l4}, {l2, l4, l5}} {{l2, l3, l5}, {l1, l3, l5}} {l1, l3, l4} {{l3, l5}, {l1, l3}} {{l2, l3, l4}, {l1, l2, l3, l4}} {l1, l3, l5} {{l1, l5}, {l2, l5}} {{l2, l3, l4}, {l1, l3, l4}} {l1, l4, l5} {{l2, l3}, {l2, l3, l5}} {{l2, l3}, {l1, l2, l3}} {l2, l3, l4} {{l1, l3, l4}, {l1, l3, l4, l5}} {{l2, l4, l5}, {l1, l4, l5}} {l2, l3, l5} {{l1, l2, l4}, {l1, l2, l4, l5}} {{l4, l5}, {l1, l4, l5}} {l2, l4, l5} {{l1, l2, l3, l4}, {l2, l3, l4, l5}} {{l2, l5}, {l1, l5}} {l3, l4, l5} {{l1, l2, l3}, {l2, l3, l5}} {{l2, l4}, {l1, l4}} {l1, l2, l3, l4} {{l1, l3, l4}, {l1, l3, l4, l5}} {{l2, l3, l4, l5}, L(SDG)} {l1, l2, l3, l5} {{l1, l2, l4}, {l1, l2, l4, l5}} {{l2, l3, l4, l5}, {l1, l3, l4, l5}} {l1, l2, l4, l5} {{l1, l2, l3, l4}, {l2, l3, l4, l5}} {{l2, l3, l5}, {l1, l2, l3, l5}} {l1, l3, l4, l5} {{l1, l2, l3}, {l2, l3, l5}} {{l2, l3, l4}, {l1, l2, l3, l4}} {l2, l3, l4, l5} {{l1, l2, l3, l4}, L(SDG)} {{l2, l4, l5}, {l1, l4, l5}} Table 14: Subint(L)(K) and Subun(L)(K) with respect to Tables 12 and 13 L(K) Subint(L)(K) Subun(L)(K) ϕ {ϕ,L(SDG)} {ϕ} L(SDG) {ϕ,L(SDG)} {L(SDG)} {l1} {ϕ,L(SDG)} {ϕ, {l2, l3}} {l2} {ϕ,L(SDG)} {{l5}, {l1, l4, l5}} {l3} {ϕ,L(SDG)} {ϕ, {l1, l4}} {l4} {ϕ,L(SDG)} {ϕ, {l2, l3}} {l5} {ϕ,L(SDG)} {ϕ, {l2}} {l1, l2} {ϕ,L(SDG)} {{l5}, L(SDG)} {l1, l3} {ϕ,L(SDG)} {ϕ, {l1, l2, l3, l4}} {l1, l4} {ϕ,L(SDG)} {{l3}, {l2, l3}} {l1, l5} {ϕ,L(SDG)} {ϕ, {l2, l3}} {l2, l3} {ϕ,L(SDG)} {{l1, l4, l5}} {l2, l4} {ϕ,L(SDG)} {L(SDG), {l5}} {l2, l5} {ϕ,L(SDG)} {{l1, l2, l4, l5}, {l5}} {l3, l4} {ϕ,L(SDG)} {ϕ, {l1, l2, l3, l4}} {l3, l5} {ϕ,L(SDG)} {{l1, l2, l4}, ϕ} Continued on next page A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 17 of 27 Table 14 – Continued from previous page L(K) Subint(L)(K) Subun(L)(K) {l4, l5} {ϕ,L(SDG)} {ϕ, {l2, l3}} {l1, l2, l3} {ϕ,L(SDG)} {L(SDG), {l1, l4, l5}} {l1, l2, l4} {ϕ,L(SDG)} {{l3, l5}, L(SDG)} {l1, l2, l5} {ϕ,L(SDG)} {{l5}, L(SDG)} {l1, l3, l4} {ϕ,L(SDG)} {{l1, l2, l3, l4}, {l3}} {l1, l3, l5} {ϕ,L(SDG)} {ϕ, {l1, l2, l3, l4}} {l1, l4, l5} {ϕ,L(SDG)} {{l2, l3}} {l2, l3, l4} {ϕ,L(SDG)} {L(SDG), {l1, l4, l5}} {l2, l3, l5} {ϕ,L(SDG)} {{l1, l2, l4, l5}, {l1, l4, l5}} {l2, l4, l5} {ϕ,L(SDG)} {L(SDG), {l5}} {l3, l4, l5} {ϕ,L(SDG)} {ϕ, {l1, l2, l3, l4}} {l1, l2, l3, l4} {ϕ,L(SDG)} {L(SDG), {l1, l3, l4, l5}} {l1, l2, l3, l5} {ϕ,L(SDG)} {L(SDG), {l1, l4, l5}} {l1, l2, l4, l5} {ϕ,L(SDG)} {L(SDG), {l2, l3, l5}} {l1, l3, l4, l5} {ϕ,L(SDG)} {{l3}, {l1, l2, l3, l4}} {l2, l3, l4, l5} {ϕ,L(SDG)} {L(SDG){l1, l4, l5}} Table 15: Bt(L)(K) with respect to Table 14 L(K) Bt(L)(K) ϕ {ϕ, {l5}, L(SDG)} L(SDG) {{l1, l2, l3, l4}, L(SDG)} {l1} {ϕ, {l5}, L(SDG)} {l2} {{l1, l4}, {l4, l5}, {l4}, L(SDG)} {l3} {{l1}, {l5}, ϕ, L(SDG)} {l4} {{l3}, {l3, l5}, L(SDG)} {l5} {{l2}, {l2, l5}, L(SDG)} {l1, l2} {{l1, l4}, {l4, l5}, {l4}, L(SDG)} {l1, l3} {{l1}, {l5}, ϕ, L(SDG)} {l1, l4} {{l3}, {l3, l5}, L(SDG)} {l1, l5} {{l2}, {l2, l5}, L(SDG)} {l2, l3} {{l1, l4}, {l1, l4, l5}, L(SDG)} {l2, l4} {{l1, l3, l4}, {l3, l4, l5}, {l3, l4}, L(SDG)} {l2, l5} {{l1, l2, l4}, {l2, l4, l5}, {l2, l4}, L(SDG)} {l3, l4} {{l1, l3}, {l3, l5}, {l3}, L(SDG)} {l3, l5} {{l1, l2}, {l2, l5}, {l2}, L(SDG)} {l4, l5} {{l2, l3}, {l2, l3, l5}, L(SDG)} {l1, l2, l3} {{l1, l4}, {l1, l4, l5}, L(SDG)} {l1, l2, l4} {{l1, l3, l4}, {l3, l4, l5}, {l3, l4}, L(SDG)} Continued on next page A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 18 of 27 Table 15 – Continued from previous page L(K) Bt(L)(K) {l1, l2, l5} {{l1, l2, l4}, {l2, l4, l5}, {l2, l4}, L(SDG)} {l1, l3, l4} {{l3, l5}, {l1, l3}, {l3}, L(SDG)} {l1, l3, l5} {{l1, l5}, {l2, l5}, {l5}, L(SDG)} {l1, l4, l5} {{l2, l3}, {l2, l3, l5}, L(SDG)} {l2, l3, l4} {{l1, l3, l4}, {l1, l3, l4, l5}, L(SDG)} {l2, l3, l5} {{l1, l2, l4}, {l1, l2, l4, l5}, L(SDG)} {l2, l4, l5} {{l1, l2, l3, l4}, {l2, l3, l4, l5}, {l2, l3, l4}, L(SDG)} {l3, l4, l5} {{l1, l2, l3}, {l2, l3, l5}, {l2, l3}, L(SDG)} {l1, l2, l3, l4} {{l1, l3, l4}, {l1, l3, l4, l5}, L(SDG)} {l1, l2, l3, l5} {{l1, l2, l4}, {l1, l2, l4, l5}, L(SDG)} {l1, l2, l4, l5} {{l1, l2, l3, l4}, {l2, l3, l4, l5}, {l2, l3, l4}, L(SDG)} {l1, l3, l4, l5} {{l1, l2, l3}, {l2, l3, l5}, {l2, l3}, L(SDG)} {l2, l3, l4, l5} {{l1, l2, l3, l4}, L(SDG)} Table 16: Bn(L)(K) with respect to Table 14 L(K) Bn(L)(K) ϕ {ϕ, {l1}, L(SDG)} L(SDG) {{l2, l3, l4, l5}, L(SDG)} {l1} {{l1, l3}, {l2, l3}, {l3}, L(SDG)} {l2} {{l5}, {l1, l5}, L(SDG)} {l3} {{l4}, {l1, l4}, L(SDG)} {l4} {{l1}, {l2}, ϕ, L(SDG)} {l5} {ϕ, {l1}, L(SDG)} {l1, l2} {{l1, l3, l5}, {l2, l3, l5}, {l3, l5}, L(SDG)} {l1, l3} {{l1, l3, l4}, {l2, l3, l4}{l3, l4}, L(SDG)} {l1, l4} {{l2, l3}, {l1, l2, l3}, L(SDG)} {l1, l5} {{l1, l3}, {l2, l3}, {l3}, L(SDG)} {l2, l3} {{l4, l5}, {l1, l4, l5}, L(SDG)} {l2, l4} {{l2, l5}, {l1, l5}, {l5}, L(SDG)} {l2, l5} {{l5}, {l1, l5}, L(SDG)} {l3, l4} {{l1, l4}, {l2, l4}, {l4}, L(SDG)} {l3, l5} {{l4}, {l1, l4}, L(SDG)} {l4, l5} {{l1}, {l2}, ϕ, L(SDG)} {l1, l2, l3} {{l1, l3, l4, l5}, {l2, l3, l4, l5}, {l3, l4, l5}, L(SDG)} {l1, l2, l4} {{l1, l2, l3, l5}, {l2, l3, l5}, L(SDG)} {l1, l2, l5} {{l2, l3, l5}, {l1, l3, l5}, {l3, l5}, L(SDG)} {l1, l3, l4} {{l2, l3, l4}, {l1, l2, l3, l4}, L(SDG)} {l1, l3, l5} {{l2, l3, l4}, {l1, l3, l4}, {l3, l4}, L(SDG)} Continued on next page A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 19 of 27 Table 16 – Continued from previous page L(K) Bn(L)(K) {l1, l4, l5} {{l2, l3}, {l1, l2, l3}, L(SDG)} {l2, l3, l4} {{l2, l4, l5}, {l1, l4, l5}, {l4, l5}, L(SDG)} {l2, l3, l5} {{l4, l5}, {l1, l4, l5}, L(SDG)} {l2, l4, l5} {{l2, l5}, {l1, l5}, {l5}, L(SDG)} {l3, l4, l5} {{l2, l4}, {l1, l4}, {l4}, L(SDG)} {l1, l2, l3, l4} {{l2, l3, l4, l5}, L(SDG)} {l1, l2, l3, l5} {{l2, l3, l4, l5}, {l1, l3, l4, l5}, {l3, l4, l5}, L(SDG)} {l1, l2, l4, l5} {{l2, l3, l5}, {l1, l2, l3, l5}, L(SDG)} {l1, l3, l4, l5} {{l2, l3, l4}, {l1, l2, l3, l4}, L(SDG)} {l2, l3, l4, l5} {{l2, l4, l5}, {l1, l4, l5}, {l4, l5}, L(SDG)} Table 17: Bint(L)(K) and Bun(L)(K) with respect to Table 15 L(K) Bint(L)(K) Bun(L)(K) ϕ {ϕ,L(SDG)} {ϕ,L(SDG)} L(SDG) {ϕ,L(SDG)} {L(SDG)} {l1} {ϕ,L(SDG)} {ϕ, {l2, l3}, L(SDG)} {l2} {ϕ,L(SDG)} {{l5}, {l1, l4, l5}, L(SDG)} {l3} {ϕ,L(SDG)} {ϕ, {l1, l4}, L(SDG)} {l4} {ϕ,L(SDG)} {ϕ, {l2, l3}, L(SDG)} {l5} {ϕ,L(SDG)} {ϕ, {l2}, L(SDG)} {l1, l2} {ϕ,L(SDG)} {{l5}, L(SDG)} {l1, l3} {ϕ,L(SDG)} {ϕ, {l1, l2, l3, l4}, L(SDG)} {l1, l4} {ϕ,L(SDG)} {{l3}, {l2, l3}, L(SDG)} {l1, l5} {ϕ,L(SDG)} {ϕ, {l2, l3}, L(SDG)} {l2, l3} {ϕ,L(SDG)} {{l1, l4, l5}, L(SDG)} {l2, l4} {ϕ,L(SDG)} {L(SDG), {l5}} {l2, l5} {ϕ,L(SDG)} {{l1, l2, l4, l5}, {l5}, L(SDG)} {l3, l4} {ϕ,L(SDG)} {ϕ, {l1, l2, l3, l4}, L(SDG)} {l3, l5} {ϕ,L(SDG)} {{l1, l2, l4}, ϕ, L(SDG)} {l4, l5} {ϕ,L(SDG)} {ϕ, {l2, l3}, L(SDG)} {l1, l2, l3} {ϕ,L(SDG)} {L(SDG), {l1, l4, l5}} {l1, l2, l4} {ϕ,L(SDG)} {{l3, l5}, L(SDG)} {l1, l2, l5} {ϕ,L(SDG)} {{l5}, L(SDG)} {l1, l3, l4} {ϕ,L(SDG)} {{l1, l2, l3, l4}, {l3}, L(SDG)} {l1, l3, l5} {ϕ,L(SDG)} {ϕ, {l1, l2, l3, l4}, L(SDG)} {l1, l4, l5} {ϕ,L(SDG)} {{l2, l3}, L(SDG)} {l2, l3, l4} {ϕ,L(SDG)} {L(SDG), {l1, l4, l5}} Continued on next page A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 20 of 27 Table 17 – Continued from previous page L(K) Bint(L)(K) Bun(L)(K) {l2, l3, l5} {ϕ,L(SDG)} {{l1, l2, l4, l5}, {l1, l4, l5}, L(SDG)} {l2, l4, l5} {ϕ,L(SDG)} {L(SDG), {l5}} {l3, l4, l5} {ϕ,L(SDG)} {ϕ, {l1, l2, l3, l4}, L(SDG)} {l1, l2, l3, l4} {ϕ,L(SDG)} {L(SDG), {l1, l3, l4, l5}} {l1, l2, l3, l5} {ϕ,L(SDG)} {L(SDG), {l1, l4, l5}} {l1, l2, l4, l5} {ϕ,L(SDG)} {L(SDG), {l2, l3, l5}} {l1, l3, l4, l5} {ϕ,L(SDG)} {{l3}, {l1, l2, l3, l4}, L(SDG)} {l2, l3, l4, l5} {ϕ,L(SDG)} {L(SDG){l1, l4, l5}} Table 18: τNt(L)(K) with respect to Table 16 L(K) τNt(L)(K) ϕ {ϕ, {l5}, L(SDG)} L(SDG) {ϕ, {l1, l2, l3, l4}, L(SDG)} {l1} {ϕ, {l5}, L(SDG)} {l2} {ϕ, {l1, l4}, {l4, l5}, {l4}, {l1, l4, l5}, L(SDG)} {l3} {{l1}, {l5}, {l1, l5}, ϕ, L(SDG)} {l4} {ϕ, {l3}, {l3, l5}, L(SDG)} {l5} {ϕ, {l2}, {l2, l5}, L(SDG)} {l1, l2} {ϕ, {l1, l4}, {l4, l5}, {l4}, {l1, l4, l5}, L(SDG)} {l1, l3} {{l1}, {l5}, {l1, l5}, ϕ, L(SDG)} {l1, l4} {ϕ, {l3}, {l3, l5}, L(SDG)} {l1, l5} {ϕ, {l2}, {l2, l5}, L(SDG)} {l2, l3} {ϕ, {l1, l4}, {l1, l4, l5}, L(SDG)} {l2, l4} {ϕ, {l1, l3, l4}, {l3, l4, l5}, {l3, l4}, {l1, l3, l4, l5}, L(SDG)} {l2, l5} {ϕ, {l1, l2, l4}, {l2, l4, l5}, {l2, l4}, {l1, l2, l4, l5}, L(SDG)} {l3, l4} {ϕ, {l1, l3}, {l3, l5}, {l3}, {l1, l3, l5}, L(SDG)} {l3, l5} {ϕ, {l1, l2}, {l2, l5}, {l2}, {l1, l2, l5}, L(SDG)} {l4, l5} {ϕ, {l2, l3}, {l2, l3, l5}, L(SDG)} {l1, l2, l3} {ϕ, {l1, l4}, {l1, l4, l5}, L(SDG)} {l1, l2, l4} {ϕ, {l1, l3, l4}, {l3, l4, l5}, {l3, l4}, {l1, l3, l4, l5}, L(SDG)} {l1, l2, l5} {ϕ, {l1, l2, l4}, {l2, l4, l5}, {l2, l4}, {l1, l2, l4, l5}, L(SDG)} {l1, l3, l4} {ϕ, {l3, l5}, {l1, l3}, {l3}, {l1, l3, l5}, L(SDG)} {l1, l3, l5} {ϕ, {l1, l5}, {l2, l5}, {l5}, {l1, l2, l5}, L(SDG)} {l1, l4, l5} {ϕ, {l2, l3}, {l2, l3, l5}, L(SDG)} {l2, l3, l4} {ϕ, {l1, l3, l4}, {l1, l3, l4, l5}, L(SDG)} {l2, l3, l5} {ϕ, {l1, l2, l4}, {l1, l2, l4, l5}, L(SDG)} {l2, l4, l5} {ϕ, {l1, l2, l3, l4}, {l2, l3, l4, l5}, {l2, l3, l4}, L(SDG)} {l3, l4, l5} {ϕ, {l1, l2, l3}, {l2, l3, l5}, {l2, l3}, {l1, l2, l3, l5}, L(SDG)} Continued on next page A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 21 of 27 Table 18 – Continued from previous page L(K) τNt(L)(K) {l1, l2, l3, l4} {ϕ, {l1, l3, l4}, {l1, l3, l4, l5}, L(SDG)} {l1, l2, l3, l5} {ϕ, {l1, l2, l4}, {l1, l2, l4, l5}, L(SDG)} {l1, l2, l4, l5} {ϕ, {l1, l2, l3, l4}, {l2, l3, l4, l5}, {l2, l3, l4}, L(SDG)} {l1, l3, l4, l5} {ϕ, {l1, l2, l3}, {l2, l3, l5}, {l2, l3}, {l1, l2, l3, l5}, L(SDG)} {l2, l3, l4, l5} {ϕ, {l1, l2, l3, l4}, L(SDG)} Table 19: τNn(L)(K) with respect to Table 17 L(K) τNn(L)(K) ϕ {ϕ, {l1}, L(SDG)} L(SDG) {ϕ, {l2, l3, l4, l5}, L(SDG)} {l1} {ϕ, {l1, l3}, {l2, l3}, {l3}, {l1, l2, l3}, L(SDG)} {l2} {ϕ, {l5}, {l1, l5}, L(SDG)} {l3} {ϕ, {l4}, {l1, l4}, L(SDG)} {l4} {ϕ, {l1}, {l2}, {l1, l2}, ϕ, L(SDG)} {l5} {ϕ, {l1}, L(SDG)} {l1, l2} {ϕ, {l1, l3, l5}, {l2, l3, l5}, {l3, l5}, {l1, l2, l3, l5}, L(SDG)} {l1, l3} {ϕ, {l1, l3, l4}, {l2, l3, l4}{l3, l4}, {l1, l2, l3, l4}, L(SDG)} {l1, l4} {ϕ, {l2, l3}, {l1, l2, l3}, L(SDG)} {l1, l5} {ϕ, {l1, l3}, {l2, l3}, {l3}, {l1, l2, l3}, L(SDG)} {l2, l3} {ϕ, {l4, l5}, {l1, l4, l5}, L(SDG)} {l2, l4} {ϕ, {l2, l5}, {l1, l5}, {l5}, {l1, l2, l5}, L(SDG)} {l2, l5} {ϕ, {l5}, {l1, l5}, L(SDG)} {l3, l4} {ϕ, {l1, l4}, {l2, l4}, {l4}, {l1, l2, l4}, L(SDG)} {l3, l5} {ϕ, {l4}, {l1, l4}, L(SDG)} {l4, l5} {{l1}, {l2}, {l1, l2}, ϕ, L(SDG)} {l1, l2, l3} {ϕ, {l1, l3, l4, l5}, {l2, l3, l4, l5}, {l3, l4, l5}, L(SDG)} {l1, l2, l4} {ϕ, {l1, l2, l3, l5}, {l2, l3, l5}, L(SDG)} {l1, l2, l5} {ϕ, {l2, l3, l5}, {l1, l3, l5}, {l3, l5}, {l1, l2, l3, l5}, L(SDG)} {l1, l3, l4} {ϕ, {l2, l3, l4}, {l1, l2, l3, l4}, L(SDG)} {l1, l3, l5} {ϕ, {l2, l3, l4}, {l1, l3, l4}, {l3, l4}, {l1, l2, l3, l4}, L(SDG)} {l1, l4, l5} {ϕ, {l2, l3}, {l1, l2, l3}, L(SDG)} {l2, l3, l4} {ϕ, {l2, l4, l5}, {l1, l4, l5}, {l4, l5}, {l1, l2, l4, l5}, L(SDG)} {l2, l3, l5} {ϕ, {l4, l5}, {l1, l4, l5}, L(SDG)} {l2, l4, l5} {ϕ, {l2, l5}, {l1, l5}, {l5}, {l1, l2, l5}, L(SDG)} {l3, l4, l5} {ϕ, {l2, l4}, {l1, l4}, {l4}, {l1, l2, l4}, L(SDG)} {l1, l2, l3, l4} {ϕ, {l2, l3, l4, l5}, L(SDG)} {l1, l2, l3, l5} {ϕ, {l2, l3, l4, l5}, {l1, l3, l4, l5}, {l3, l4, l5}, L(SDG)} {l1, l2, l4, l5} {ϕ, {l2, l3, l5}, {l1, l2, l3, l5}, L(SDG)} Continued on next page A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 22 of 27 Table 19 – Continued from previous page L(K) τNn(L)(K) {l1, l3, l4, l5} {ϕ, {l2, l3, l4}, {l1, l2, l3, l4}, L(SDG)} {l2, l3, l4, l5} {ϕ, {l2, l4, l5}, {l1, l4, l5}, {l4, l5}, {l1, l2, l4, l5}, L(SDG)} Table 20: τint(L)(K) and τun(L)(K) with respect to Table 18 L(K) τint(L)(K) τun(L)(K) ϕ {ϕ,L(SDG)} {ϕ,L(SDG)} L(SDG) {ϕ,L(SDG)} ϕ, {L(SDG)} {l1} {ϕ,L(SDG)} {ϕ, {l2, l3}, L(SDG)} {l2} {ϕ,L(SDG)} {ϕ, {l5}, {l1, l4, l5}, L(SDG)} {l3} {ϕ,L(SDG)} {ϕ, {l1, l4}, L(SDG)} {l4} {ϕ,L(SDG)} {ϕ, {l2, l3}, L(SDG)} {l5} {ϕ,L(SDG)} {ϕ, {l2}, L(SDG)} {l1, l2} {ϕ,L(SDG)} {ϕ, {l5}, L(SDG)} {l1, l3} {ϕ,L(SDG)} {ϕ, {l1, l2, l3, l4}, L(SDG)} {l1, l4} {ϕ,L(SDG)} {ϕ, {l3}, {l2, l3}, L(SDG)} {l1, l5} {ϕ,L(SDG)} {ϕ, {l2, l3}, L(SDG)} {l2, l3} {ϕ,L(SDG)} {ϕ, {l1, l4, l5}, L(SDG)} {l2, l4} {ϕ,L(SDG)} {ϕ,L(SDG), {l5}} {l2, l5} {ϕ,L(SDG)} {ϕ, {l1, l2, l4, l5}, {l5}, L(SDG)} {l3, l4} {ϕ,L(SDG)} {ϕ, {l1, l2, l3, l4}, L(SDG)} {l3, l5} {ϕ,L(SDG)} {{l1, l2, l4}, ϕ, L(SDG)} {l4, l5} {ϕ,L(SDG)} {ϕ, {l2, l3}, L(SDG)} {l1, l2, l3} {ϕ,L(SDG)} {ϕ,L(SDG), {l1, l4, l5}} {l1, l2, l4} {ϕ,L(SDG)} {ϕ, {l3, l5}, L(SDG)} {l1, l2, l5} {ϕ,L(SDG)} {ϕ, {l5}, L(SDG)} {l1, l3, l4} {ϕ,L(SDG)} {ϕ, {l1, l2, l3, l4}, {l3}, L(SDG)} {l1, l3, l5} {ϕ,L(SDG)} {ϕ, {l1, l2, l3, l4}, L(SDG)} {l1, l4, l5} {ϕ,L(SDG)} {ϕ, {l2, l3}, L(SDG)} {l2, l3, l4} {ϕ,L(SDG)} {ϕ,L(SDG), {l1, l4, l5}} {l2, l3, l5} {ϕ,L(SDG)} {ϕ, {l1, l2, l4, l5}, {l1, l4, l5}, L(SDG)} {l2, l4, l5} {ϕ,L(SDG)} {ϕ,L(SDG), {l5}} {l3, l4, l5} {ϕ,L(SDG)} {ϕ, {l1, l2, l3, l4}, L(SDG)} {l1, l2, l3, l4} {ϕ,L(SDG)} {ϕ,L(SDG), {l1, l3, l4, l5}} {l1, l2, l3, l5} {ϕ,L(SDG)} {ϕ,L(SDG), {l1, l4, l5}} {l1, l2, l4, l5} {ϕ,L(SDG)} {ϕ,L(SDG), {l2, l3, l5}} {l1, l3, l4, l5} {ϕ,L(SDG)} {ϕ, {l3}, {l1, l2, l3, l4}, L(SDG)} {l2, l3, l4, l5} {ϕ,L(SDG)} {ϕ,L(SDG){l1, l4, l5}} Remark 3. Figure 2 shows that the reversal of stocks is not achieved and this has been A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 23 of 27 clearly made clear through examples 4 and 5. Figure 2: Comparison between results. Remark 4. The relation between τt, τn, τint, τun and T0 and T1 (i) Necessary and sufficient conditions for To. A topological space is T0 if for every pair of distinct vertices x ̸= y, there ex- ists an open set containing one but not the other. Firstly, for τt: The subbase is {LONt(K), UPNt(K)}, and the topology is generated from out-neighbourhoods Nt(v). The condition for T0 : τt is T0 iff for all u ≠ v,Nt(u) \ {v} ̸= Nt(v) \ {u} is neces- sary but not sufficient. Sufficient condition: τt is T0 iff for all u ̸= v, there exists w such that w ∈ Nt(u) △ Nt(v) and this difference yields an open set containing one but not the other. In the airline example (Fig. 2), Table 19 shows τt is not discrete, but likely fails T0 for some vertices, e.g. if Nt(u) = Nt(v) then they are topologically indistinguishable. Secondly, for τn: the condition for T0 : τn is T0 iff for all u ̸= v,Nn(u) ̸= Nn(v), or the topology distinguishes them through the in-links. Third, for τint : Nint(v) = Nt(v) ∩Nn(v). If Nint(v) = ϕ for all v, then LONint(K) is either ϕ or L(SDG) depending on K, and UPNint(K) is ϕ or L(SDG) similarly. Then τint = {ϕ,L} is trivial. The trivial topology is not To (cannot separate any points). So, τint is T0 only if Nint is nonempty enough to distinguish vertices, which is rare. Finally, for τun : Nun(v) = Nt(v) ∪ Nn(v). τun is T0 iff Nun(u) ̸= Nun(v) for all u ̸= v. (ii) Can any of τt, τn, τint, τun be T1? A space is T1 if for every u ≠ v, there is an open set containing u but not v and another containing v but not u. τt is T1 iff the digraph has no edges (discrete topology). τn is T1 iff there are no edges (discrete). τint is almost never T1. τun is T1 iff Nun(v) = {v} for all v iff the graph has no edges (discrete). In general, T0 is achievable for τt, τn, τun under certain distinguishing neighbourhood conditions, but τint is almost never T0. T1 is impossible for any nontrivial digraph for all four topologies, they are T1 only in the discrete case (not edges). A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 24 of 27 Remark 5. Analysis exploring whether continuous self maps on the constructed spaces satisfy any known fixed-point theorems. (i) Defining a ”j-Neighbourhood Preserving” Mapping Definition (j-Continuous Map): Let (V, τj) be a topological space constructed from a simple directed graph SDG(V ) using the j-neighbourhood system j ∈ {t, n, int, un}). A function f : V → V is called j-continuous if it is continuous with respect to the topology τj, i.e., for every open set U ∈ τj, its pre-image f−1(U) is also open in τj. Definition (j-Neighbourhood Preserving Map): A function f : V → V is j- neighbourhood Preserving if for every vertex v ∈ V , the image of the j-neighbourhood is contained in the j-neighbourhood of the image: f(Nj(v)) ⊆ Nj(f(v)). This is a direct analogue of a continuous map in a neighbourhood space: points ”close” to v (in Nj)) are mapped to points ”close” to f(v). (ii) Connecting to known fixed-point Theorems Classical fixed-point theorems (Brouwer, Schauder) require topological structure like compactness and convexity, which our finite discrete spaces possess trivially (they are compact), but they lack the convex structure. The most relevant theorem for finite topological spaces is a direct consequence of the Lefschetz Fixed-Point Theorem. (iii) Fixed- Point Theorem for j-Continuous Maps Theorem 1 (Fixed point in acyclic graphs): Let SDG(V ) be a finite, acyclic directed graph. Let f : V → V be a j-continuous map for j ∈ {t, n}. Then f has a fixed point. Proof: Acyclicity implies a ”source” or ”sink”: In a finite acyclic digraph, there exists at least one source (vertex with Nn(v) = ϕ) and one sink (vertex with Nt(v) = ϕ). Topology of τt in an acyclic graph: Consider the out-neighbourhood topology τt. For a sink vertex s,Nt(s) = ϕ. Therefore, LONt({S}) = {x ∈ V : Nt(x) ⊆ {s}}. Since s is a sink, Nt(s) = ϕ ⊆ {s}, so s ∈ LONt({s}). In fact, for many acyclic graphs, the set of sinks form minimal open sets. The fixed-point argument: Let S be the set of sink vertices. This set is open in τt because for each sink s, the singleton {s} might be open, or S is a union of such minimal open sets. A τt-continuous map f must map sinks to sinks. Suppose s is a sink and f(s) is not a sink. Then there is an edge f(s) → w. The set U = {v ∈ V : w /∈ Nt(v)} is an open neighbourhood of s (since Nt(s) = ϕ,w /∈ Nt(s)). By continuity, f−1(U) is open and contains s. But f(s) /∈ U , which is a contradiction regarding the pre-image. Thus, f(s) must be a sink. Since f maps the finite set S to itself, by the pigeonhole principle (or the finite fixed point property), f has a fixed point within S. Theorem 2 (Fixed point in strongly connected graphs with j-contractive property): Let SDG(V ) be a finite, strongly connected directed graph. Let f : V → V be a j-continuous map that is j-contracting for j = un, meaning: f(Nun(v)) ⊆ Nun(f(v)) for all v ∈ V where Nun(v) ̸= {v}. Then f has a fixed point. A. Abushaaban, A. El-Atik, O. Embaby / Eur. J. Pure Appl. Math, 18 (4) (2025), 6694 25 of 27 proof: Strong connectivity and τun : In a strongly connected graph, the union neigh- bourhood topology τun is highly non-trivial. The space (V, τun) is connected. Fur- thermore, because every very vertex is reachable from every other, the minimal open sets (given by LONun({v})) are not singletons (unless the graph is a single vertex). The contracting property implies a unique ”center”: A contracting map on a finite, connected topological space that is ”sphere-like” (in the sense of the neighbourhood structure) must have a unique fixed point. The argument is analogous to the Branch fixed-point theorem but for finite metric spaces or ultra-metric spaces. Here, the ”metric” is the graph distance in the underlying undirected graph of Nun. Fixed point via minimal closed set: Consider the family F of non-empty, closed subsets C of (V, τun) such that f(C) ⊆ C. This family is non-empty (as V itself is in it). Since V is finite, there is a minimal such set C0. If C0 has more than one point, the con- tracting property and the connectivity of C0 (in the subspace topology) would force f to map C0 to a proper subset of itself that is also closed and f -invariant, contradicting minimality. Therefore, C0 must be a singleton {v0}, and hence f(v0) = v0. 5. Conclusions and future work This study successfully introduced and investigated new types of topological spaces in simple directed graphs, leveraging the concept of j-neighbourhoods. We defined four distinct j-neighbourhoods (outside, inside, intersection, and union) and subsequently used them to establish j-lower and j-upper approximations for subgraphs. The detailed exam- ples and tables provided throughout the document illustrate the practical application of these definitions and approximations. We also rigorously proved several propositions that highlight the fundamental properties of these approximations concerning set operations such as inclusion, union, intersection, and complementation. This theoretical framework offers a more refined approach to analyzing graph structures and their inherent relation- ships. For future work, this research can be extended in several directions: Firstly, further ex- ploration of topological properties: Investigate additional topological properties induced by j-neighbourhoods, such as connectedness, compactness, and separation axioms, to gain a deeper understanding of these new topological spaces. Secondly, applications in diverse fields: Explore the applicability of these new topological spaces in other complex net- works beyond air travel, such as social networks, biological networks, and communication networks, to model and analyze their structural characteristics and dynamics. Thirdly, weighted and fuzzy graphs: Extend the definitions of j-neighbourhoods and approxima- tions to weighted graphs, where edges have associated values, and fuzzy graphs, where relationships are not sharply defined, to enhance the model’s capacity to represent real- world complexities. Fourthly, algorithmic development: Develop efficient algorithms for computing J-neighbourhoods, lower and upper approximations, and the induced topo- logical spaces for large-scale graphs. 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