Abstract
For a given graph [Formula: see text], by [Formula: see text] and [Formula: see text] denote the order of the largest component and the number of connected components of [Formula: see text], respectively. The rupture degree of [Formula: see text] is an important combinatorial parameters, which defined as [Formula: see text]. Clearly, the smaller the value [Formula: see text] is, the better the connectivity of graph [Formula: see text] 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.
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