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# Extremal graphs with given order and the rupture degree.

Computers & Mathematics with Applications 01/2010; 60:1706-1710. DOI: 10.1016/j.camwa.2010.07.001

Source: DBLP

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**ABSTRACT:**The toughness of a graph G is defined as the largest real number t such that deletion of any s points from G results in a graph which is either connected or else has at most s/t components. Clearly, every hamiltonian graph is 1-tough. Conversely, we conjecture that for some t0, every t0-tough graph is hamiltonian. Since a square of a k-connected graph is always k-tough, a proof of this conjecture with t0 = 2 would imply Fleischner's theorem (the square of a block is hamiltonian). We construct an infinite family of ()-tough nonhamiltonian graphs.Discrete Mathematics 05/2006; · 0.58 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In assessing the “vulnerability” of a graph one determines the extent to which the graph retains certain properties after the removal of a number of vertices and/or edges. Four measures of vulnerability to vertex removal are compared for classes of graphs with edge densities ranging from that of trees to that of the complete graph.JCMCC. The Journal of Combinatorial Mathematics and Combinatorial Computing. 01/1987; 1. -
##### Article: Rupture degree of graphs.

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**ABSTRACT:**We introduce a new graph parameter, the rupture degree. The rupture degree for a complete graph Kn is defined as 1−n, and the rupture degree for an incomplete connected graph G is defined by r(G)=max{ω(G−X)−|X|−m(G−X):X⊂V(G), ω(G−X)>1}, where ω(G−X) is the number of components of G−X and m(G−X) is the order of a largest component of G−X. It is shown that this parameter can be used to measure the vulnerability of networks. Rupture degrees of several specific classes of graphs are determined. Formulas for the rupture degree of join graphs and some bounds of the rupture degree are given. We also obtain some Nordhaus–Gaddum type results for the rupture degree.International Journal of Computer Mathematics 01/2005; 82:793-803. · 0.54 Impact Factor

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