Abstract
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 for connected graphs with a fixed number of vertices. We discuss such pairs in the case when the dimension is relatively large. As a corollary, we completely determine the pairs for connected graphs with a small number of vertices. We also study the possible pairs for connected chordal graphs.
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