Consider a simple graph G. A labeling w :The goal is to obtain a total vertex product-irregular labeling that minimizes the maximum label. This minimum value is called the total vertex product irregularity strength and denoted tvps(G). In this paper we provide some general lower and upper bounds, as well as exact values for chosen families of graphs.
Let G = (V, E) be a graph of order n. A distance magic labeling of G is a bijection : V → {1, . . . , n} for which there exists a positive integer k such thatWe introduce a natural subclass of distance magic graphs. For this class we show that it is closed for the direct product with regular graphs and closed as a second factor for lexicographic product with regular graphs. In addition, we characterize distance magic graphs among direct product of two cycles.
a b s t r a c tConsider a simple graph G with no isolated edges and at most one isolated vertex. AThe goal is to obtain a product-irregular labeling that minimizes the maximum label. This minimum value is called the product irregularity strength. The analogous concept of irregularity strength, with sums in place of products, has been introduced by Chartrand et al. and investigated by many authors.
Let G = (V, E) be a graph of order n. A closed distance magic labeling of G is a bijection ℓ : V (G) → {1, . . . , n} for which there exists a positive integer k suchis the closed neighborhood of v. We consider the closed distance magic graphs in the algebraic context. In particular we analyze the relations between the closed distance magic labelings and the spectra of graphs. These results are then applied to the strong product of graphs with complete graph or cycle and to the circulant graphs. We end with a number theoretic problem whose solution results in another family of closed distance magic graphs somewhat related to the strong product.
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