A labeling of a graph is a mapping that carries some set of graph elements into numbers (usually positive integers). An (a, d)-edge-antimagic total labeling of a graph, with p vertices and q edges, is a one-to-one mapping that takes the vertices and edges into the integers 1, 2, . . . , p + q, so that the sums of the label on the edges and the labels of their end vertices form an arithmetic progression starting at a and having difference d. Such a labeling is called super if the p smallest possible labels appear at the vertices. This thesis deals with the existence of super (a, d)-edge-antimagic total labelings of regular graphs and disconnected graphs. We prove that every even regular graph and every odd regular graph, with a 1- factor, admits a super (a, 1)-edge-antimagic total labeling. We study the super (a, 2)- edge-antimagic total labelings of disconnected graphs and present some necessary conditions for the existence of (a, d)-edge-antimagic total labelings for d even. The thesis is also devoted to the study of edge-antimagicness of trees. We use the connection between graceful labelings and edge-antimagic labelings for generating large classes of edge-antimagic total trees from smaller graceful trees.