Discovering Irregularities from Computer Networks by Topological Mapping
Abstract
:1. Introduction
2. Literature Review
3. Research Methodology
3.1. Objectives
3.2. Significance
3.3. Method
4. Experimental Results
4.1. Main Results
4.1.1. Bridge Graph Gr (Ps, v) over Path
4.1.2. Theorem 1
4.2. Main Results
4.2.1. Bridge Graph Gr (Cs, v) over Cycle
4.2.2. Theorem 2
4.3. Main Results
4.3.1. Bridge Graph Gr (Ks, v) over Complete Graph
4.3.2. Theorem 3
4.4. Main Results of Honeycomb Network
4.4.1. Honeycomb Graph
4.4.2. Theorem 4
4.5. Main Results of Regular Hexagonal Cell Network
4.5.1. Regular Hexagonal Cells Network
4.5.2. Theorem 5
4.6. Main Results
4.6.1. Sierpinski Network Graph
4.6.2. Theorem 6
4.7. Main Results
4.7.1. Sierpinski Network S(n, k)
4.7.2. Theorem 7
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Sr. No. | Title of Research Paper | Year | Networks Solved | Invariants Used | Results |
---|---|---|---|---|---|
1 | Topological Properties Of Degree-Based Invariants Via M-Polynomial Approach | 2022 | Hexagonal Networks | Zagreb Indices, Randi’C, Product Connectivity Gourava Index and their Forms | Give valuable information about the molecular structure or network and applications in QSPR & QSAR. |
2 | Contraharmonic Quadratic Index Of Certain Nanostar Dendrimers | 2022 | Dendrimer Nanostars | Contraharmonic-Quadratic Index and Quadratic-Contraharmonic Index | computed the CQ index for some standard graphs |
3 | Some Results On The Sombor Indices of Graphs | 2021 | Degree-Regular Graph/Network | The Sombor Index, The Reduced Sombor Index and the Average Sombor Index | Establishing inequalities related to the aforementioned three graph invariants and proving a recently proposed conjecture concerning the sombor index |
4 | Some Basic Properties of Sombor Indices | 2021 | Regular Graph or Network | Vertex-Degree-Based (VDB) Molecular Structure Descriptors (Sombor Index and its Reduced Form) | Any reduced VDB index can be viewed as a reduced sombor-type index |
5 | Analysis Of Dendrimer Generation By Sombor Indices | 2021 | Dendrimers Generation Networks | Sombor Index and Reduced Sombor Index | Computed sombor indices for phosphorus-containing dendrimers & types of dendrimers. |
6 | Sombor Index of Some Nanostructures | 2021 | Nanostructures | Sombor Index | Computed explicit formulae for sombor index of 2D-lattice, nanotube, and nanotorus |
7 | Polynomials And General Degree-Based Topological Indices of Generalized Sierpinski Networks | 2021 | Sierpinski Networks | Connectivity Polynomials Such As m-Polynomial, Zagreb Polynomials, Forgotten Polynomial, (A, Β)-Zagreb Index and Several Other General Indices | These facts can be Physicochemical properties of the molecules modeled on the S(k, n) networks can be forecasted using the results. |
8 | The Calculations of Topological Indices on Certain Networks | 2021 | Hexagonal Networks | ABC Index, AZI Index, GA Index, The Multiplicative Version Of Ordinary First Zagreb Index, The Second Multiplicative Zagreb Index, and Zagreb Index | Calculating the correlation index provides potential help for scholars to study networks characteristics better. for further work, if the corresponding networks are replaced by other networks |
9 | Discovering Irregularities from Computer Networks by Topological Mapping | 2022 | Bridge Networs, Hexagonal Networks, Honeycomb Networks and Sierpinski Networks | Irregularity Sombor Index | Finding Sharp upper bounds, lower bounds and irregularities |
ε | ε(du, dv) | de | ε(du, de) | Recurrence |
---|---|---|---|---|
ε1 | ε(1, 2) | 1 | ε(1, 1) | R |
ε2 | ε(2, 2) | 2 | ε(2, 2) | 3r + 2 |
ε3 | ε(2, 3) | 3 | ε(2, 3) | R |
ε4 | ε(3, 3) | 4 | ε(3, 4) | r − 3 |
ε | ε(du, dv) | de | ε(du, de) | Recurrence |
---|---|---|---|---|
ε1 | ε(2, 2) | 2 | ε(2, 2) | rs − 2r |
ε2 | ε(2, 3) | 3 | ε(2, 3) | 4 |
ε3 | ε(2, 4) | 4 | ε(2, 4) | 2r − 4 |
ε4 | ε(3, 4) | 5 | ε(3, 5) | 2 |
ε5 | ε(4, 4) | 6 | ε(4, 6) | r − 3 |
ε | ε(du, dv) | De | ε(du, de) | Recurrence |
---|---|---|---|---|
ε1 | ε(4, 5) | 7 | ε(4, 7) | 2 |
ε2 | ε(4, S−1) | S + 1 | ε(4, s+1) | 2 |
ε3 | ε(5, 5) | 8 | ε(5, 8) | r − 2 |
ε4 | ε(5, S−1) | S + 2 | ε(5, s+2) | r − 2 |
ε5 | ε(S−1, S−1) | 2s − 4 | ε(s−1, 2s−4) | [rs(r − 1) − 2(r + 1)]/2 |
Ε | ε(du, dv) | de | ε(du, de) | Recurrence |
---|---|---|---|---|
ε1 | ε(5, 5) | 8 | ε(5, 8) | 6 |
ε2 | ε(5, 7) | 10 | ε(5,10) | 12(n − 1) |
ε3 | ε(7, 9) | 14 | ε(7, 14) | 6(n − 1) |
ε4 | ε(9, 9) | 16 | ε(9, 16) | 9n2 − 21n + 12 |
Ε | ε(du, dv) | de | ε(du, de) | Recurrence |
---|---|---|---|---|
ε1 | ε(2, 2) | 2 | ε(2, 2) | 2n + 4 |
ε2 | ε(2, 3) | 3 | ε(2, 3) | 4m + 4n + 4 |
ε3 | ε(3,3) | 4 | ε(3, 4) | 6 mn + m − 5n − 4 |
Ε | ε(du, dv) | De | ε(du, de) | Recurrence |
---|---|---|---|---|
ε1 | ε(2, 4) | 4 | ε(2, 4) | 6 |
ε2 | ε(4, 4) | 6 | ε(4, 6) | 3n − 6 |
ε | ε(du, dv) | De | ε(du, de) | Recurrence |
---|---|---|---|---|
ε1 | ε(2, k) | k | ε(2, k) | 2k |
ε2 | ε(3, 3) | 4 | ε(3, 4) | (kn+1 − 5k)/2 |
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Hamid, K.; Waseem Iqbal, M.; Abbas, Q.; Arif, M.; Brezulianu, A.; Geman, O. Discovering Irregularities from Computer Networks by Topological Mapping. Appl. Sci. 2022, 12, 12051. https://doi.org/10.3390/app122312051
Hamid K, Waseem Iqbal M, Abbas Q, Arif M, Brezulianu A, Geman O. Discovering Irregularities from Computer Networks by Topological Mapping. Applied Sciences. 2022; 12(23):12051. https://doi.org/10.3390/app122312051
Chicago/Turabian StyleHamid, Khalid, Muhammad Waseem Iqbal, Qaiser Abbas, Muhammad Arif, Adrian Brezulianu, and Oana Geman. 2022. "Discovering Irregularities from Computer Networks by Topological Mapping" Applied Sciences 12, no. 23: 12051. https://doi.org/10.3390/app122312051
APA StyleHamid, K., Waseem Iqbal, M., Abbas, Q., Arif, M., Brezulianu, A., & Geman, O. (2022). Discovering Irregularities from Computer Networks by Topological Mapping. Applied Sciences, 12(23), 12051. https://doi.org/10.3390/app122312051