### On additive vertex labelings

#### Abstract

In a quite general sense, additive vertex labelings are those functions that assign nonnegative integers to the vertices of a graph and the weight of each edge is obtained by adding the labels of its end-vertices. In this work we study one of these functions, called harmonious labeling. We calculate the number of non-isomorphic harmoniously labeled graphs with *n* edges and at most n vertices. We present harmonious labelings for some families of graphs that include certain unicyclic graphs obtained via the corona product. In addition, we prove that all *n*-cell snake polyiamonds are harmonious; this type of graph is obtained via edge amalgamation of n copies of the cycle *C*_{3} in such a way that each copy of this cycle shares at most two edges with other copies. Moreover, we use the edge-switching technique on the cycle *C*_{4t }to generate unicyclic graphs with another type of additive vertex labeling, called strongly felicitous, which has a solid bond with the harmonious labeling.

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PDFDOI: http://dx.doi.org/10.19184/ijc.2020.4.1.5

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