Transitional Values of Graphs

M.A. Johnson introduced a way to describe a chemical transformation by labeling each vertex and edge of a graph G with one of integers - 1, 0, 1. The labeling of G so obtained is called a transitional labeling of G. The value of a transitional labeling of G is defined as the minimum of the number of negative elements of G and the number of positive elements of G. We introduce a new invariant of a graph G called the transitional value of G, defined as the maximum value among all the values of the transitional labelings of G. This parameter provides a measure of the structural changes that occur in a chemical transformation represented by a transitional labeling of G. We determine the transitional values of some families of graphs and characterize the transitional labelings of maximum value of complete graphs.