Edge ideals of finite simple graphs $G$ on $n$ vertices are the ideals $I(G)$ of the polynomial ring $S$ in $n$ variables generated by the quadratic monomials associated with the edges of $G$. In this paper, we consider the possible pairs of dimensions and depths of $S/I(G)$ for connected graphs with a fixed number of vertices. We discuss such pairs in the case where dimension is relatively large. As a corollary, we completely determine the pairs for connected graphs with small number of vertices. We also study the possible pairs for connected chordal graphs.