Extremal Graphs on the Rupture Degree

For a given graph G = (V,E), by tau(G) and omega(G) denote the order of the largest component and the number of connected components of G, respectively. The rupture degree of G is an important combinatorial parameters, which defined as r(G)=max {omega(G - X) -vertical bar X vertical bar- tau(G - X) : X subset of V (G), omega(G - X) > 1}. Clearly, the smaller the value r(G) is, the better the connectivity of graph G is. In this paper, the tree structures with minimum rupture degree be discussed, and the extremal graphs on the rupture degree in terms of graph girth are also characterized.