Integer-Magic Spectra Of Amalgamations Of Stars And Cycles

For any positive integer k, a graph G = (V, E) is said to be ZZ k-magic if there exists a labeling l : E(G) −→ ZZ k − {0} such that the induced vertex set labeling l : V (G) −→ ZZ k defined by l(v) = ∑ { l(uv) : uv ∈ E(G) } is a constant map. For a given graph G, the set of all h ∈ ZZ + for which G is ZZ h-magic is called the integer-magic spectrum of G and is denoted by IM(G). In this paper, we will determine the integer-magic spectra of the graphs which are formed by the amalgamation of stars and cycles. In particular, we will provide examples of graphs that for a given n > 2, they are not h-magic for all values of 2 ≤ h ≤ n.