#### Semester

Summer

#### Date of Graduation

2005

#### Document Type

Dissertation

#### Degree Type

PhD

#### College

Eberly College of Arts and Sciences

#### Department

Mathematics

#### Committee Chair

Sam B. Nadler, Jr.

#### Abstract

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.

#### Recommended Citation

Simon Romero, Likin C., "The hyperspace graph of connected subgraphs" (2005). *Graduate Theses, Dissertations, and Problem Reports*. 2321.

https://researchrepository.wvu.edu/etd/2321