On the degrees of a strongly vertex-magic graph

Let G=(V,E)G=(V,E) be a finite graph, where |V|=n⩾2|V|=n⩾2 and |E|=e⩾1|E|=e⩾1. A vertex-magic total labeling is a bijection λλ from V∪EV∪E to the set of consecutive integers {1,2,…,n+e}{1,2,…,n+e} with the property that for every v∈Vv∈V, λ(v)+∑w∈N(v)λ(vw)=h for some constant h  . Such a labeling is strong if λ(V)={1,2,…,n}λ(V)={1,2,…,n}. In this paper, we prove first that the minimum degree of a strongly vertex-magic graph is at least two. Next, we show that if 2e⩾10n2-6n+1, then the minimum degree of a strongly vertex-magic graph is at least three. Further, we obtain upper and lower bounds of any vertex degree in terms of n and e. As a consequence we show that a strongly vertex-magic graph is maximally edge-connected and hamiltonian if the number of edges is large enough. Finally, we prove that semi-regular bipartite graphs are not strongly vertex-magic graphs, and we provide strongly vertex-magic total labeling of certain families of circulant graphs.