Given a connected graph G, the hyperspace graph of connected subgraphs C (G) is defined. The graph C (G) is such that every vertex represents a connected subgraph of G. It is shown that every connected graph G has a unique graph C (G). A characterization of a path, a cycle and the 3-star by their corresponding hyperspace graphs of connected subgraphs is shown. A special geometric representation R (G) of C (G) in an euclidean space is presented. A set P (G) is constructed based on R (G). When G is a topological tree, P (G) and the hyperspace of subcontinua of G are homeomorphic.; Given a graph G, the size of G is the cardinality of the edge set. A special kind of subgraphs of C (G) is studied; given a non-negative integer n, the n-th size level of G, denoted by Qn (G) is defined. This graph is the induced graph in C (G) of all the connected subgraphs of G with size n. Relations between G, the graphs Qn (G) and C (G) are analyzed.
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