This book creates a new door to a whole new field for research. It was originated by creating analogous ideas from topology to graph theory. Nevertheless, everyone that has taken an undergraduate course in graph theory will be able to understand this work. Given a connected graph, we can construct a new connected graph whose vertices represent all the connected subgraphs of the original graph. This defines the Hyperspace Graph of Connected Subgraphs. One of the objectives is finding relations between the properties of a graph and its corresponding Hyperspace Graph of Connected Subgraphs. In this book, we explore just a few of this relations and give just a little tiny view of this vast field of open questions and problems. This book is intended to invite mathematicians young and old, novice and experienced to explore and expand the ideas presented. The work within this pages is not a end, a solution, but a beginning, a question and a generator for new results.

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