RESIDUAL CLOSENESS FOR HELM AND SUNFLOWER GRAPHS

Abstract

Vulnerability is an important concept in network analysis related with the ability of the network to avoid intentional attacks or disruption when a failure is produced in some of its components. Often enough, the network is modeled as an undirected and unweighted graph in which vertices represent the processing elements and edges represent the communication channel between them. Dierent measures for graph vulnerability have been introduced so far to study dierent aspects of the graph behavior after removal of vertices or links such as connectivity, toughness, scattering number, binding number and integrity. In this paper, we consider residual closeness which is a new characteristic for graph vulnerability. Residual closeness is a more sensitive vulnerability measure than the other measures of vulnerability. We obtain exact values for closeness, vertex residual closeness VRC and normalized vertex residual closeness NVRC for some wheel related graphs namely helm and sun ower.

Keywords

• network vulnerability,closeness,network design and communication,stabil- ity,communication network,Helm graph,Sun ower graph.

References

1. Aytac,A. and Odabas,Z.N., (2011), Residual Closeness of Wheels and Related Networks, International Journal of Foundations of Computer Science, 22(5), pp.1229-1240.
2. Aytac,A. and Berberler,Z.N.O., (2017), Network Robustness and Residual Closeness, RAIRO Opera- tion Research, (accepted).
3. Barefoot,C.A., Entringer,R., and Swart,H., (1987), Vulnerability in graphs– a comparative survey, J.
4. Combin. Math. Combin. Comput.1, pp.13-22. Chvatal,V., (1973), Tough graphs and Hamiltonian circuits, Discrete Math. 5, pp.215-228.
5. Dangalchev,Ch., (2011), Residual closeness and generalized closeness, International Journal of Founda- tons of Computer Science, 22(8), pp.1939-1948.
6. Dangalchev,Ch., (2006), Residual Closeness in Networks, Physica A, 365, pp.556-564.
7. Gutman,I., (1998), Distance of thorny graphs, Publ. Inst. Math. (Beograd), 63, pp.31-36.
8. Gallian,J.A., (2008), A dynamic survey of graph labeling, Elect. Jour. Combin., 15 DS6.

9. Javaid,I. and Shokat,S., (2008), On the partition dimension of some wheel related graphs, Journal of
10. Prime Research in Mathematics 4, pp.154-164. Jung,H.A., (1978), On a class of posets and the corresponding comparability graphs, Journal of
11. Combinatorial Theory, Series B 24(2), pp.125-133. Odabas,Z.N. and Aytac,A. (2013), Residual closeness in cycles and related networks, Fund. Inform., (3), pp.297-307.
12. Stevanovi´c,D., (2001) Hosoya polynomial of composite graphs, Discrete Mathematics,235, pp.237-244.
13. West,D.B., (2001), Introduction to Graph Theory, Prentice Hall, NJ.
14. Woodall,D.R., (1973) The binding number of a graph and its Anderson number, J. Combin. Theory Ser. B 15, pp.225-255.