Super Connectivity of Erdős-Rényi Graphs
Abstract
:1. Introduction
2. Main Results
- (i)
- If , then:
- (ii)
- If , then:
- (i)
- If , then:
- (ii)
- If , then:
3. Concluding Remarks
Funding
Acknowledgments
Conflicts of Interest
References
- Albert, R.; Jeong, H.; Barabási, A.-L. Error and attack tolerance of complex networks. Nature 2000, 406, 378–382. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Shang, Y. Vulnerability of networks: Fractional percolation on random graphs. Phys. Rev. E 2014, 89, 012813. [Google Scholar] [CrossRef] [PubMed]
- Bauer, D.; Boesch, F.; Suffel, C.; Tindell, R. Connectivity extremal problems and the design of reliable probabilistic networks. In The Theory and Application of Graphs; Wiley: New York, NY, USA, 1981; pp. 45–54. [Google Scholar]
- Boesch, F.T. Synthesis of reliable networks: A survey. IEEE Trans. Reliab. 1986, 35, 240–246. [Google Scholar] [CrossRef]
- Chang, J.-M.; Chen, X.-R.; Yang, J.-S.; Wu, R.-Y. Locally exchanged twisted cubes: connectivity and super connectivity. Inform. Process. Lett. 2016, 116, 460–466. [Google Scholar] [CrossRef]
- Ekinci, G.B.; Gauci, J.B. On the reliability of generalized Petersen graphs. Discrete Appl. Math. 2019, 252, 2–9. [Google Scholar]
- Ekinci, G.B.; Kirlangi, A. Super connectivity of Kronecker product of complete bipartite graphs and complete graphs. Discrete Math. 2016, 339, 1950–1953. [Google Scholar]
- Lin, C.-K.; Huang, H.-M.; Hsu, L.-H. The super connectivity of the pancake graphs and the super laceability of the star graphs. Theory Comput. Sci. 2005, 339, 257–271. [Google Scholar] [CrossRef] [Green Version]
- Yang, M. Super connectivity of balanced hypercubes. Appl. Math. Comput. 2012, 219, 970–975. [Google Scholar] [CrossRef]
- Zhu, Q.; Zhang, J.; Li, L.L. The h-extra connectivity and h-extra conditional diagnosabillity of Bubble-sort star graphs. Discrete Appl. Math. 2018, 251, 322–333. [Google Scholar] [CrossRef]
- Frieze, A.; Karoński, M. Introduction to Random Graphs; Cambridge University Press: New York, NY, USA, 2016. [Google Scholar]
- Bollobás, B. Random Graphs; Cambridge University Press: New York, NY, USA, 2001. [Google Scholar]
- Janson, S.; Łuczak, T.; Rucinski, A. Random Graphs; Wiley: New York, NY, USA, 2000. [Google Scholar]
- Erdős, P.; Rényi, A. On the strength of connectedness of a random graph. Acta Math. Acad. Sci. Hungar. 1961, 12, 261–267. [Google Scholar] [CrossRef]
- Boesch, F.; Tindell, R. Circulants and their connectivities. J. Graph Theory 1984, 8, 487–499. [Google Scholar] [CrossRef]
- Bollobás, B.; Frieze, A. On matchings and hamiltonian cycles in random graphs. Ann. Discrete Math. 1985, 28, 23–46. [Google Scholar]
- Harary, F. Conditional connectivity. Networks 1983, 13, 347–357. [Google Scholar] [CrossRef]
- Federico, L.; van der Hofstad, R. Critical window for connectivity in the configuration model. Comb. Prob. Comput. 2017, 26, 660–680. [Google Scholar] [CrossRef]
- Fountoulakis, N.; Müller, T. Law of large numbers for the largest component in a hyperbolic model of complex networks. Ann. Appl. Prob. 2018, 28, 607–650. [Google Scholar] [CrossRef] [Green Version]
- Iyer, S.K. The random connection model: Connectivity, edge lengths, and degree distributions. Rand. Struct. Algor. 2018, 52, 283–300. [Google Scholar] [CrossRef]
- Zhao, J.; Yağan, O.; Gligor, V. On connectivity and robustness in random intersection graphs. IEEE Trans. Autom. Contr. 2017, 62, 2121–2136. [Google Scholar] [CrossRef]
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Shang, Y. Super Connectivity of Erdős-Rényi Graphs. Mathematics 2019, 7, 267. https://doi.org/10.3390/math7030267
Shang Y. Super Connectivity of Erdős-Rényi Graphs. Mathematics. 2019; 7(3):267. https://doi.org/10.3390/math7030267
Chicago/Turabian StyleShang, Yilun. 2019. "Super Connectivity of Erdős-Rényi Graphs" Mathematics 7, no. 3: 267. https://doi.org/10.3390/math7030267
APA StyleShang, Y. (2019). Super Connectivity of Erdős-Rényi Graphs. Mathematics, 7(3), 267. https://doi.org/10.3390/math7030267