A graph with p vertices is said to be strongly multiplicative if its vertices can be labelled 1, 2, . . . , p so that the values on the edges, obtained as the product of the labels of their end vertices, are all distinct. In this paper, we study structural properties of strongly multiplicative graphs. We show that all graphs in some classes, including all trees, are strongly multiplicative, and consider the question of the maximum number of edges in a strongly multiplicative graph of a given order.
A ( p , +graph G is said to be (k, &arithmetic if its vertices can be assigned distinct nonnegative integers so that the values of the edges, obtained as the sums of the numbers assigned to their end vertices, can be arranged in the arithmetic progression k, kIn this paper we initiate a study on the structures of finite (k, &arithmetic graphs.
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