# On The Average Lower Independence Number of Some Graphs

## Abstract

In a communication network, several vulnerability measures areused to determine the resistance of the network to disruption of operation after the failure of certain stations or communication links. If the communication network is modeled as a simple, undirected, connected and unweighted graph G, then average lower independence number of a graph G can be considered

as a measure of graph vulnerability and is dened by

iav (G) = 1 jV (G)j Σ v2V (G) iv(G)

where iv(G) is the minimum cardinality of a maximal independent set that contains v. In this paper, I consider the average lower independence number of the thorn graphs of special graphs.

## References

[1] A. Aytac and T. Turaci. The average lower independence number of total graphs, Bull. Int. Math. Virtual Inst., 2(1)(2012), 17-27.

[2] K.S. Bagga and L.W. Beineke. Survey of Integrity, Discrete Applied Math., 37-38 (1992), 13-28.

[3] C.A Barefoot, R. Entringer and H. Swart. Vulnerability in Graphs-A Comparative Survey, J. Comb. Math. Comb. Comput., 1 (1987), 13-22.

[4] M. Blidia, M. Chellali and F. Maffray. On Average Lower Independence and Domination Numbers in Graphs, Discrete Math., 295(1-3)(2005), 1-11.

[5] J. A. Bondy and U. S. R. Murty. Graph theory with applications, American Elsevier Publishing Co. Inc. New York (1976).

[6] V. Chvtal, Tough graphs and Hamiltonian circuits. Discrete Math., 306(10-11)(2006), 910-917.

[7] Cozzens, M.B., D. Moazzami, S. Stueckle, The tenacity of a Graph, In Y Alavi and A. Schwenk (Eds.): Graph Theory, Combinatorics, and Algorithms (pp. 1111-1122) Wiley, New

York, (1995).

[8] I. Gutman. Distance Of Thorny Graphs, Publ. de L'Institute Math., N.S., 63(77)(1998), 31-36.

[9] F. Harary and F. Buckley. Distance in Graphs, Addison-Wesley Publishing Company (1989).

[10] J. Haviland. Independent domination in regular graphs, Discrete Math., 143(1-3)(1995), 275-280.

[11] M. A. Henning. Trees with equal average domination and independent domination numbers, Ars Combinatoria, 71(2004), 305-318.

[12] L. Sun and J.Wang. An Upper Bound for the Independent Domination Number, J. of Comb. Theory series B, 76(2)(1999), 240-246.

[13] H. Tuncel, T. Turaci and B. Coskun. On the average lower domination number and some results of complementary prisms and graph join, J. Adv. Res. Appl. Math., 7(1)(2015), 52-61.

[14] T. Turaci and M. Okten. Vulnerability of Mycielski graphs via residual closeness, Ars Combinatoria, 118(2015), 419-427.

[2] K.S. Bagga and L.W. Beineke. Survey of Integrity, Discrete Applied Math., 37-38 (1992), 13-28.

[3] C.A Barefoot, R. Entringer and H. Swart. Vulnerability in Graphs-A Comparative Survey, J. Comb. Math. Comb. Comput., 1 (1987), 13-22.

[4] M. Blidia, M. Chellali and F. Maffray. On Average Lower Independence and Domination Numbers in Graphs, Discrete Math., 295(1-3)(2005), 1-11.

[5] J. A. Bondy and U. S. R. Murty. Graph theory with applications, American Elsevier Publishing Co. Inc. New York (1976).

[6] V. Chvtal, Tough graphs and Hamiltonian circuits. Discrete Math., 306(10-11)(2006), 910-917.

[7] Cozzens, M.B., D. Moazzami, S. Stueckle, The tenacity of a Graph, In Y Alavi and A. Schwenk (Eds.): Graph Theory, Combinatorics, and Algorithms (pp. 1111-1122) Wiley, New

York, (1995).

[8] I. Gutman. Distance Of Thorny Graphs, Publ. de L'Institute Math., N.S., 63(77)(1998), 31-36.

[9] F. Harary and F. Buckley. Distance in Graphs, Addison-Wesley Publishing Company (1989).

[10] J. Haviland. Independent domination in regular graphs, Discrete Math., 143(1-3)(1995), 275-280.

[11] M. A. Henning. Trees with equal average domination and independent domination numbers, Ars Combinatoria, 71(2004), 305-318.

[12] L. Sun and J.Wang. An Upper Bound for the Independent Domination Number, J. of Comb. Theory series B, 76(2)(1999), 240-246.

[13] H. Tuncel, T. Turaci and B. Coskun. On the average lower domination number and some results of complementary prisms and graph join, J. Adv. Res. Appl. Math., 7(1)(2015), 52-61.

[14] T. Turaci and M. Okten. Vulnerability of Mycielski graphs via residual closeness, Ars Combinatoria, 118(2015), 419-427.

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## Published

2017-03-02

## Issue

## Section

Чланци