Abstract
We study the relationship between the smallest positive vertex degrees and the (vertex) connectivity in 1-dimensional line-of-sight (LoS) networks. We show that these networks differ from their higher dimensional counterparts. While a strong relationship exists between the presence of certain low degree vertices and the connectivity of d-dimensional line-of-sight networks, for \(d>1\), the relationship is much weaker in the 1-dimensional case. In particular, the absence of isolated vertices is not sufficient to guarantee connectivity.
Mr. Gu worked on this project during his final year as an undergraduate at the University of Liverpool and, later, in the months leading to the start of his PhD. M. Gu is currently a PhD student in Liverpool.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Obviously \(\mathrm{\textbf{E}}X_l = 0\) for \(l > 2\omega \), and it may well be that \(\mathrm{\textbf{E}}X_l \rightarrow 0\) for \(l<2\omega \) but \(l = l(n) \rightarrow \infty \). We do not investigate that region of values, related to the maximum degree of the LoS network, in this report.
References
Buckley, F., Harary, F.: Distance in Graphs. Addison-Wesley, Boston (1990)
Bollobás, B.: Random Graphs. Cambridge Studies in Advanced Mathematics, vol. 73, 2nd edn. Cambridge University Press, Cambridge (2001)
Chung, K.L.: A Course in Probability Theory. Academic Press, Cambridge (1974)
Devroye, L., Farczadi, L.: Connectivity for line-of-sight networks in higher dimensions. Discret. Math. Theor. Comput. Sci. 15(2), 71–86 (2013)
Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173. Springer, Heidelberg (2000)
Frieze, A., Kleinberg, J., Ravi, R., Debany, W.: Line-of-sight networks. Comb. Probab. Comput. 18(1–2), 145–163 (2009)
Gilbert, E.N.: Random graphs. Ann. Math. Stat. 30, 1141–1144 (1959)
Hall, B., Paulitsch, M., Driscoll, K.: FlexRay BRAIN fusion a FlexRay-based braided ring availability integrity network. SAE Trans. 116, 460–473 (2007)
Watts, D.J., Strogatz, S.H.: Collective dynamics of “small-world” networks. Nature 393(4), 440–442 (1998)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Gu, H., Zito, M. (2023). Minimum Degree and Connectivity in 1-Dimensional Line-of-Sight Networks. In: Georgiou, K., Kranakis, E. (eds) Algorithmics of Wireless Networks. ALGOWIN 2023. Lecture Notes in Computer Science, vol 14061. Springer, Cham. https://doi.org/10.1007/978-3-031-48882-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-031-48882-5_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-48881-8
Online ISBN: 978-3-031-48882-5
eBook Packages: Computer ScienceComputer Science (R0)