آمار
| تعداد نشریات | 6 |
| تعداد شمارهها | 125 |
| تعداد مقالات | 1,585 |
| تعداد مشاهده مقاله | 1,755,116 |
| تعداد دریافت فایل اصل مقاله | 1,636,670 |
The monophonic pebbling number of neural networks with generalized algorithm and their applications | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 6، دوره 11، شماره 3، آذر 2026، صفحه 827-842 اصل مقاله (2 M) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.29481.2017 | ||
| نویسندگان | ||
| K.C. Kavitha* ؛ S. Jagatheswari* | ||
| Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology Vellore - 632014, Tamil Nadu, India | ||
| چکیده | ||
| Consider a graph $\sigma$(V, E) with nodes V and edges E is a connected graph with some pebbles scattered over its nodes V. By removal of two pebbles from one node and placing one pebble to an adjacent node is a pebbling move. A monophonic pebbling number, $\lambda_{M}(\sigma, v)$, of a node v of a graph $\sigma$ is the least number $m$ such that minimum of one pebble could be shifted to v by a sequence of pebbling shifts for any distribution of $\lambda_{M}(\sigma, v)$ pebbles on the nodes of $\sigma$ using monophonic path. A link between the nodes x and y is an x-y path which consists of no chords and is monophonic. The monophonic pebbling number of a graph $\sigma$ is the highest $\lambda_{M}(\sigma, v)$ among all the nodes notated as $\lambda_{M}(\sigma)$. For the first time, we calculate the monophonic pebbling number on families of neural networks such as probabilistic neural networks(PNNs), convolutional neural networks(CVNNs), modular neural networks(MNNs), generalized regression neural networks(GRNNs) and Hopfield neural networks(HNNs) and discuss their applications. We give the generalized algorithm to find the monophonic pebbling number of any graph $\sigma$. | ||
| کلیدواژهها | ||
| monophonic pebbling number؛ (PNNs)؛ (CVNNs)؛ (MNNs)؛ (GRNNs) and (HNNs) | ||
| مراجع | ||
|
[1] G. Auda and M. Kamel, Modular neural networks: a survey, Int. J. Neural Syst. 9 (1999), no. 2, 129–51. https://doi.org/10.1142/S0129065799000125
[2] J. Binwal, R. Devi, and B. Singh, Mathematical modelling and simulation of fingerprint analysis using graph isomorphism, domination, and graph pebbling, Adv. Appl. Discrete Math. 39 (2023), no. 2, 259–284. https://doi.org/10.17654/0974165823052
[3] F.R.K. Chung, Pebbling in hypercubes, SIAM J. Discrete Math. 2 (1989), no. 4, 467–72. https://doi.org/10.1137/0402041
[4] M. Defferrard, X. Bresson, and P. Vandergheynst, Convolutional neural networks on graphs with fast localized spectral filtering, Advances in neural information processing systems (Barcelona, Spain) (D.D. Lee, U. von Luxburg, R. Garnett, M. Sugiyama, and I. Guyon, eds.), 30th Conference on Neural Information Processing Systems, Curran Associates Inc., 57 Morehouse Lane, Red Hook, NY, United States, September 2016. [5] D.S. Herscovici and A.W. Higgins, The pebbling number of $C_5\times C_5$, Discrete Math. 187 (1998), no. 1–3, 123–135. https://doi.org/10.1016/S0012-365X(97)00229-X
[6] G. Hurlbert, A survey of graph pebbling, Congr. Numer. 139 (1999), 41–64.
[7] G. Hurlbert, Graph pebbling, pp. 1428–1449, Chapman and Hall/CRC, Kalamazoo, 2013.
[8] G. Isaak, M. Prudente, and J.M. Marcinik III, A pebbling game on powers of paths, Commun. Number Theory Comb. Theory 4, Article No: 1.
[9] K.C. Kavitha and S. Jagatheswari, Monophonic rubbling number of some standard graphs, Heliyon 10 (2024), no. 11, Article ID: e31679. https://doi.org/10.1016/j.heliyon.2024.e31679
[10] A. Khan, S. Hayat, Y. Zhong, A. Arif, L. Zada, and M. Fang, Computational and topological properties of neural networks by means of graph-theoretic parameters, Alex. Eng. J. 66 (2023), 957–977. https://doi.org/10.1016/j.aej.2022.11.001
[11] J.B. Liu, M.K. Shafiq, H. Ali, A. Naseem, N. Maryam, and S.S. Asghar, Topological indices of m th chain silicate graphs, Mathematics 7 (2019), no. 1, Article ID: 42. https://doi.org/10.3390/math7010042
[12] A. Loeffler, R. Zhu, J. Hochstetter, M. Li, K. Fu, A. Diaz-Alvarez, T. Nakayama, J.M. Shine, and Z. Kuncic, Topological properties of neuromorphic nanowire networks, Front. Neurosci. 14 (2020), Article ID: 184. https://doi.org/10.3389/fnins.2020.00184
[13] A. Lourdusamy, I. Dhivviyanandam, and S. Kither Iammal, Monophonic pebbling number and t-pebbling number of some graphs, AKCE Int. J. Graphs Comb. 19 (2022), no. 2, 108–111. https://doi.org/10.1080/09728600.2022.2072789
[14] A. Lourdusamy, I. Dhivviyanandam, and S. Kither Iammal, Monophonic pebbling number of some standard graphs, South East Asian J. Math. Math. Sci. 21 (2022), 177–182.
[15] A. Lourdusamy, I. Dhivviyanandam, and S. Kither Iammal, Monophonic pebbling number of some families of cycles, Discrete Math. Algorithms Appl. 16 (2023), no. 4, Article ID: 2350038. https://doi.org/10.1142/S1793830923500386
[16] A. Lourdusamy, I. Dhivviyanandam, and S. Kither Iammal, Monophonic pebbling number of some network-related graphs, J. Appl. Math. Inform. 42 (2024), no. 1, 77–83. https://doi.org/10.14317/jami.2024.077
[17] C.J. Nelson and S. Bonner, Neuronal graphs: A graph theory primer for microscopic, functional networks of neurons recorded by calcium imaging, Front. Neural Circuits 15 (2021), Article ID: 662882. https://doi.org/10.3389/fncir.2021.662882
[18] A.P. Paul and A.B. Anisha, Cover edge pebbling number for jahangir graphs $J_{1,m}, J_{2,m}, J_{3,m}, J_{4,m}$ and $J_{5,m}, 22 (2023), no. 8, 1721–1728.
[19] A. Sadiquali and P.A.P. Sudhahar, Monophonic domination in special graph structures and related properties, Int. J. Math. Anal. 11 (2017), no. 22, 1089–1102. https://doi.org/10.12988/ijma.2017.79125
[20] A.P. Santhakumaran and P. Titus, Monophonic distance in graphs, Discrete Math. Algorithms Appl. 3 (2011), no. 2, 159–169. https://doi.org/10.1142/S1793830911001176
[21] P. Sarkar, S. Mondal, N. De, and A. Pal, On topological properties of probabilistic neural network, Malaya J. Mat. 7 (2019), 612–617. https://doi.org/10.26637/MJM0704/0002
[22] L. Zhou and X. Gu, Embedding topological features into convolutional neural network salient object detection, Neural Netw. 121 (2020), 308–318. https://doi.org/10.1016/j.neunet.2019.09.009 | ||
|
آمار تعداد مشاهده مقاله: 332 تعداد دریافت فایل اصل مقاله: 552 |
||