We call a finite simple graph G = (V (G),E(G)) to be Z-semimagic if it admits an edge labeling l : E(G) ¡÷ Z {0} such that the induced vertex sum labeling l+(v) = uv∈E(G) l(uv) is constant. The constant is called a semimagic index, or an index for short, of G under the labeling l. We consider the set of all possible semimagic indices r such that G is Z-semimagic with a semimagic index r, and denote it by IZ(G). We call IZ(G) the index set of G with respect to Z. In this thesis, we decide the index set IZ(G) for G being regular graphs, complete bipartite graphs, wheel graphs and fan graphs. Also, we determine whether 0 ∈ IZ(G) for G being complete multi-partite graphs.