Öz
The fractional versions of various metric related parameters have recently gained importance due to their applications in the fields of sensor networking, robot navigation and linear optimization problems. Convex polytopes are collection of those polytopes of Euclidean space which are their convex subsets. They have key importance in the field of network designing due to their stable and resilient structure which aids optimal data transfer. The identification and removal of components (nodes) of a communication network causing abruption in its flow is of key importance for optimal data transmission. These components are referred as strong resolving neighbourhood (SRNs) in graph theory and assigning least weight to these components aids the computation of fractional strong metric dimension (FSMD). In this paper, we compute FSMD for certain convex polytopes which include $\mathbb{P}_{n}$, $\mathbb{P}_{n}^{1}$ and $\mathbb{P}_{n}^{2}$. In this regard, it is shown that for $n \geq 3$, FSMD of $\mathbb{P}_{n}$ and $\mathbb{P}_{n}^{2}$ is $n$ and $\frac{3n}{2}$, respectively. Also, FSMD of $\mathbb{P}_{n}^{1}$ is $n$ when $n$ is odd and $\frac{3n}{2}$ when $n$ is even. Finally, an application of FSMD in the context of internet connection networks is furnished.
Anahtar Kelimeler
convex polytope metric dimension fractional strong metric dimension
Kaynakça
- [1] S. Aisyah, M.I. Utoyo and L. Susilowati, On the local fractional metric dimension of corona product graphs, IOP Conference Series: Earth and Environmental Science 243, 2019.
- [2] S. Arumugam and V. Mathew, The fractional metric dimension of graphs, Discrete Math. 312, 1584-1590, 2012.
- [3] Z. Beerliova, F. Eberhard, T. Erlebach, A. Hall, M. Hoffman, M. Mihalak and L. Ram, Network discovery and verification, IEEE J. Sel. Areas Commun. 24, 2168- 2181, 2006.
- [4] G. Chartrand, I. Eroh, M.A. Johnson and O.R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math. 105, 99 - 113, 2000.
- [5] Y. M. Chu, M. F. Nadeem, M. Azeem and M. K. Siddiqui, On Sharp Bounds on Partition Dimension of Convex Polytopes, IEEE Access 8, 224781-224790, 2020.
- [6] J. Currie and O.R. Oellermann, The metric dimension and metric independence of a graph, Journal of Combinatorial Mathematics and Combinatorial Computing 39, 157 - 167, 2001.
- [7] M. Feher, S. Gosselin and O.R. Oellermann, The metric dimension of Cayley diagraph, Discrete Math. 306, 31 - 41, 2006.
- [8] F. Harary and R.A. Melter, On the metric dimension of a graph, Ars Comb. 2, 191-195, 1976.
- [9] M. Imran, S.A.H. Bokhary and A.Q. Baig, On families of convex polytopes with constant metric dimension, Comput. Math. Appl. 60 (9), 2629-2638, 2010.
- [10] F. Jamil, A. Kashif, S. Zafar, Z. Bassfer and A.M. Alanazi, Local fractional strong metric dimension of certain rotationally symmetric planer networks, IEEE Access 9, 2021.
- [11] F. Jamil, A. Kashif, S. Zafar and A. Nawaz, Fractional strong metric resolvability in graphs (submitted).
- [12] C.X. Kang, On the fractional strong metric dimension of graphs, Discrete Appl. Math. 251, 190 - 203, 2018.
- [13] C.X. Kang and E. Yi, The fractional strong metric dimension of graphs, International Conference on Combinatorial Optimization and Applications 8287, 84-95, 2013.
- [14] S. Khullar, B. Raghavchari and A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math. 70, 217 - 229, 1996.
- [15] J.B. Liu, M.K. Aslam and M. Javaid, Local fractional metric dimensions of rotationally symmetric and planar Networks, IEEE Access 8, 82404-82420, 2020.
- [16] J.B. Liu, A. Kashif, T. Rashid and M. Javaid, Fractional metric dimension of generalized Jahangir graph, Mathematics 7 , 1-10, 2019.
- [17] F. Okamoto, L. Crosse, B. Phinezy and P. Zhang, The local metric dimension of a graph, Math. Bohem. 135, 239-255, 2010.
- [18] O.R. Ollermann and J. Peters-Fransen, The strong metric dimension of graphs and diagraphs, Discrete Appl. Math. 155, 356 - 364, 2007.
- [19] H. Raza, S. Hayat and X. Pan, On the fault-tolerant metric dimension of convex polytopes, Appl. Math. Comput. 339, 172 - 185, 2018.
- [20] A. Seb¨o and E. Tannier, On metric generators of graphs, Math. Oper. Res. 29, 383-393, 2004.
- [21] P.J. Slater, Leaves of trees, Congressus Numerantium 14, 549-559, 1975.
Öz
Anahtar Kelimeler
convex polytope metric dimension fractional strong metric dimension.
Kaynakça
- [1] S. Aisyah, M.I. Utoyo and L. Susilowati, On the local fractional metric dimension of corona product graphs, IOP Conference Series: Earth and Environmental Science 243, 2019.
- [2] S. Arumugam and V. Mathew, The fractional metric dimension of graphs, Discrete Math. 312, 1584-1590, 2012.
- [3] Z. Beerliova, F. Eberhard, T. Erlebach, A. Hall, M. Hoffman, M. Mihalak and L. Ram, Network discovery and verification, IEEE J. Sel. Areas Commun. 24, 2168- 2181, 2006.
- [4] G. Chartrand, I. Eroh, M.A. Johnson and O.R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math. 105, 99 - 113, 2000.
- [5] Y. M. Chu, M. F. Nadeem, M. Azeem and M. K. Siddiqui, On Sharp Bounds on Partition Dimension of Convex Polytopes, IEEE Access 8, 224781-224790, 2020.
- [6] J. Currie and O.R. Oellermann, The metric dimension and metric independence of a graph, Journal of Combinatorial Mathematics and Combinatorial Computing 39, 157 - 167, 2001.
- [7] M. Feher, S. Gosselin and O.R. Oellermann, The metric dimension of Cayley diagraph, Discrete Math. 306, 31 - 41, 2006.
- [8] F. Harary and R.A. Melter, On the metric dimension of a graph, Ars Comb. 2, 191-195, 1976.
- [9] M. Imran, S.A.H. Bokhary and A.Q. Baig, On families of convex polytopes with constant metric dimension, Comput. Math. Appl. 60 (9), 2629-2638, 2010.
- [10] F. Jamil, A. Kashif, S. Zafar, Z. Bassfer and A.M. Alanazi, Local fractional strong metric dimension of certain rotationally symmetric planer networks, IEEE Access 9, 2021.
- [11] F. Jamil, A. Kashif, S. Zafar and A. Nawaz, Fractional strong metric resolvability in graphs (submitted).
- [12] C.X. Kang, On the fractional strong metric dimension of graphs, Discrete Appl. Math. 251, 190 - 203, 2018.
- [13] C.X. Kang and E. Yi, The fractional strong metric dimension of graphs, International Conference on Combinatorial Optimization and Applications 8287, 84-95, 2013.
- [14] S. Khullar, B. Raghavchari and A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math. 70, 217 - 229, 1996.
- [15] J.B. Liu, M.K. Aslam and M. Javaid, Local fractional metric dimensions of rotationally symmetric and planar Networks, IEEE Access 8, 82404-82420, 2020.
- [16] J.B. Liu, A. Kashif, T. Rashid and M. Javaid, Fractional metric dimension of generalized Jahangir graph, Mathematics 7 , 1-10, 2019.
- [17] F. Okamoto, L. Crosse, B. Phinezy and P. Zhang, The local metric dimension of a graph, Math. Bohem. 135, 239-255, 2010.
- [18] O.R. Ollermann and J. Peters-Fransen, The strong metric dimension of graphs and diagraphs, Discrete Appl. Math. 155, 356 - 364, 2007.
- [19] H. Raza, S. Hayat and X. Pan, On the fault-tolerant metric dimension of convex polytopes, Appl. Math. Comput. 339, 172 - 185, 2018.
- [20] A. Seb¨o and E. Tannier, On metric generators of graphs, Math. Oper. Res. 29, 383-393, 2004.
- [21] P.J. Slater, Leaves of trees, Congressus Numerantium 14, 549-559, 1975.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematik |
| Bölüm | Matematik |
| Yazarlar | |
| Erken Görünüm Tarihi | 27 Ağustos 2024 |
| Yayımlanma Tarihi | 28 Nisan 2025 |
| Yayımlandığı Sayı | Yıl 2025 |