Implicit-degrees and circumferences
In this paper, we introduce a new concept — implicit-degrees of vertices of graphs. Related to this concept, a new definition for the closure of a graph is introduced. This closure is obtained by recursively joining pairs of nonadjacent vertices whose implicit-degree sum is at least the order of the graph. A criterion for deciding whether a graph is hamiltonian from its closure is obtained. Two other important results concerning the implicit-degrees of vertices and circumferences of graphs are presented.
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