# Extremal graphs with given order and the rupture degree

**ABSTRACT** The rupture degree of an incomplete connected graph G is defined by r(G)=max{ω(G−X)−|X|−τ(G−X):X⊂V(G),ω(G−X)>1}, where ω(G−X) is the number of components of G−X and τ(G−X) is the order of a largest component of G−X. In Li and Li [5] and Li et al. (2005) [4], it was shown that the rupture degree can be well used to measure the vulnerability of networks. In this paper, the maximum and minimum networks with prescribed order and the rupture degree are obtained. Finally, we determine the maximum rupture degree graphs with given order and size.

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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; 5(3-5):215-228. DOI:10.1016/0012-365X(73)90138-6 · 0.57 Impact Factor -
##### 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 07/2005; 82(7):793-803. DOI:10.1080/00207160412331336062 · 0.72 Impact Factor -
##### Article: ON GRAPH THEORY APPLICATIONS-I