Journal of Applied Mathematics

The Average Lower Connectivity of Graphs

Ersin Aslan

Abstract

For a vertex $v$ of a graph $G$, the lower connectivity, denoted by ${s}_{v}(G)$, is the smallest number of vertices that contains $v$ and those vertices whose deletion from $G$ produces a disconnected or a trivial graph. The average lower connectivity denoted by ${\kappa }_{\text{a}\text{v}}(G)$ is the value $({\sum }_{v\in V(G)}{s}_{v}(G))/|V(G)|$. It is shown that this parameter can be used to measure the vulnerability of networks. This paper contains results on bounds for the average lower connectivity and obtains the average lower connectivity of some graphs.

Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 807834, 4 pages.

Dates
First available in Project Euclid: 2 March 2015

https://projecteuclid.org/euclid.jam/1425305527

Digital Object Identifier
doi:10.1155/2014/807834

Mathematical Reviews number (MathSciNet)
MR3170448

Zentralblatt MATH identifier
07010757

Citation

Aslan, Ersin. The Average Lower Connectivity of Graphs. J. Appl. Math. 2014 (2014), Article ID 807834, 4 pages. doi:10.1155/2014/807834. https://projecteuclid.org/euclid.jam/1425305527

References

• K. S. Bagga, L. W. Beineke, R. E. Pippert, and M. J. Lipman, “A classification scheme for vulnerability and reliability parameters of graphs,” Mathematical and Computer Modelling, vol. 17, no. 11, pp. 13–16, 1993.
• O. R. Oellermann, “Connectivity and edge-connectivity in graphs: a survey,” Congressus Numerantium, vol. 116, pp. 231–252, 1996.
• M. A. Henning, “Trees with equal average domination and inde-pendent domination numbers,” Ars Combinatoria, vol. 71, pp. 305–318, 2004.
• L. W. Beineke, O. R. Oellermann, and R. E. Pippert, “The average connectivity of a graph,” Discrete Mathematics, vol. 252, no. 1–3, pp. 31–45, 2002.
• A. Hellwig and L. Volkmann, “Maximally edge-connected and vertex-connected graphs and digraphs: a survey,” Discrete Math-ematics, vol. 308, no. 15, pp. 3265–3296, 2008.
• Y.-C. Chen, “Super connectivity of $k$-regular interconnection networks,” Applied Mathematics and Computation, vol. 217, no. 21, pp. 8489–8494, 2011.
• J. Fàbrega and M. A. Fiol, “Maximally connected digraphs,” Journal of Graph Theory, vol. 13, no. 6, pp. 657–668, 1989.
• T. Soneoka, H. Nakada, M. Imase, and C. Peyrat, “Sufficient con-ditions for maximally connected dense graphs,” Discrete Mathematics, vol. 63, no. 1, pp. 53–66, 1987.
• P. Dankelmann and O. R. Oellermann, “Bounds on the average connectivity of a graph,” Discrete Applied Mathematics, vol. 129, no. 2-3, pp. 305–318, 2003.
• W.-S. Chiue and B.-S. Shieh, “On connectivity of the Cartesian product of two graphs,” Applied Mathematics and Computation, vol. 102, no. 2-3, pp. 129–137, 1999. \endinput