Given a simple graph, consider the polynomial ring with coefficients in a field and variables identified with the edges of the graph. Given a non-empty even cardinality Eulerian subgraph and a choice of half of its edges, consider the homogeneous binomial obtained by taking the product of these edges minus the product of the remaining edges of the subgraph. We define a homogeneous ideal by taking as generators all binomials obtained in this way, varying the Eulerian subgraph and the choice of half of its edges, together with the squares of the variables of the ring. This ideal is related to the Eulerian ideal, introduced by Neves, Vaz Pinto and Villarreal. We call the corresponding quotient the Eulerian Artinian algebra associated to the graph. The goal of the present work is to study the socle of these algebras through the lens of graph theory. Our main results include a combinatorial characterization of a monomial basis of the socle, a characterization of Gorenstein Eulerian Artinian algebras in the case of bipartite graphs and the computation of the h-vector and socle degrees in the cases of a complete graph and a complete bipartite graph.