On the degrees of E-super vertex-magic graphs

Let G=(V,E) be a finite graph, where |V|=p≥2 and |E|=q≥1. A vertex-magic total labeling is a bijection f from V∪E to {1,2,…,p+q} with the property that for every u∈V, f(u)+∑v∈N(u)f(uv)=k for some constant k. Such a labeling is E-super if f(E)={1,2,…,q}. In this paper, we prove that semi-regular bipartite graphs are not E-super vertex-magic. Also we obtain an upper bound for the maximum degree Δ(G) of an E-super vertex-magic graph and we investigate upper and lower bounds of any vertex degree d of E-super vertex-magic graph.