GÖKŞEN BACAK TURAN, Meltem Ülkü ŞENOĞLU, Ferhan Nihan ALTUNDAĞ

Neighbor Rupture Degree of Some Middle Graphs

Networks have an important place in our daily lives. Internet networks, electricity networks, water networks, transportation networks, social networks and biological networks are some of the networks we run into every aspects of our lives. A network consists of centers connected by links. A network is represented when centers and connections modelled by vertices and edges, respectively. In consequence of the failure of some centers or connection lines, measurement of the resistance of the network until the communication interrupted is called vulnerability of the network. In this study, neighbor rupture degree which is a parameter that explores the vulnerability values of the resulting graphs due to the failure of some centers of a communication network and its neighboring centers becoming nonfunctional were applied to some middle graphs and neighbor rupture degree of the $M(C_{n}),$ $M(P_{n}),$ $M(K_{1,n}),$ $M(W_{n}),$ $M(P_{n}\times K_{2})$ and $M(C_{n}\times K_{2})$ have been found.

Anahtar Kelimeler:

Graph theory, Vulnerability; Neighbor rupture degree; Middle graphs

PDF

___

[1] Bondy, J.A., Murty, U.S.R. 1976. Graph theory with application. Elsevier Science Ltd/North-Holland. 264s
[2] Chavatal, V. 1973. Tough graphs and Hamiltonian circuits. Discrete Math, 5(3), 215-228.
[3] Entringer, R., Swart H. 1987. Vulnerability in Graphs- A Comparative Survey. J. Combin. Math. Combin. Comput, 1, 12-22.
[4] Cozzens, M., Moazzami, D., Stueckle, S. 1995. The Tenacity of a graph. Graph theory, combinatorics and algorithms, 1, 1-2.
[5] Jung, H. A. 1978. On a class of posets and the corresponding comparability graphs. J. Combinatorial Theory Series B, 24(2), 125-133.
[6] Li, Y., Zhang, S., Li, X. 2005. Rupture degree of graphs. International Journal of Computer Mathematics, 82(7), 793-803.
[7] Bacak-Turan, G., Kirlangic, A. 2011. Neighbor Rupture Degree and The Relations Between Other Parameters. Ars Combinatoria, 102, 333-352.
[8] Gunther, G. 1985. Neighbor connectivity in regular graphs. Discrete Applied Mathematics, 11(3), 233-243.
[9] Cozzens, M. B., Wu, S.S.Y. 1996. Vertex-neighbor Integrity of trees. Ars Combinatoria, 43, 169-180.
[10] Kirlangic, A. 2004. Graph Operations and Neighbor Integrity. Mathematica Bohemica, 129(3), 245-254.
[11] Wei, Z.T. 2003. On the reliability parameters of networks. Northwestern Polytechnical University, MSc. Thesis, 40s.
[12] Kurkcu, O.K., Aksan, H. 2016. Neighbor Toughness of graphs. Bulletin of International Mathematical Virtual Institue, 6(2), 135-141.
[13] Aslan, E. 2015. Neighbor Isolated Tenacity of Graphs. RAIRO-Theor. Inf.Appl. 49(4), 269-284.
[14] Nihei, M.2001. On the toughness of the middle graph of a graph. Ars Combinatoria, 49, 55-58.
[15] Mamut, A., Vulmar, E.2007. A Note on the Integrity of Middle Graphs. Discrete Geometry, Combinatorics and Graph Theory, Lecture Notes in Computer Science, 4381, 130-134.
[16] Odabas, Z., Aytac, A. 2012. Rupture Degree and Middle Graphs. Comptess rendus de I’Acade’mie bulgare des Sciences, 65(3), 315-322.
[17] Aytac, A., Turaci, T., Odabas, Z. 2013. On the bondage number of middle graphs. Mathematical Notes, 93(5), 795-801.
[18] Aytac, A. 2005. On the edge-tenacity of the middle graph of a graph. International Journal of Computer Mathematics, 82(5), 551-558.