Kranda, Daniel Joseph2023-11-172023-11-172011https://hdl.handle.net/10365/33250It can be shown that any symmetric (0, 1)-matrix A with tr A = 0 can be interpreted as the adjacency matrix of a simple, finite graph. The square of an adjacency matrix A2 = (si1) has the property that Sij represents the number of walks of length two from vertex i to vertex j. With this information, the motivating question behind this paper was to determine what conditions on a matrix S are needed to have S = A(G)2 for some graph G. Structural results imposed by the matrix S include detecting bipartiteness or connectedness, counting four cycles and determining plausible neighborhoods of vertices. Some characterizations will be given and the problem of when S represents several non-isomorphic graphs is also explored.NDSU policy 190.6.2https://www.ndsu.edu/fileadmin/policy/190.pdfMatrices.Graph theory.Thesis