Summary: 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 onto the integers $1, 2\cdots ,p+q$, so that the sum of the labels on the edges and the labels of their end vertices forms 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. In this paper we prove that every even regular graph and every odd regular graph with a 1-factor are super $(a,1)$-edge-antimagic total. We also introduce some constructions of non-regular super $(a,1)$-edge-antimagic total graphs.