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This paper studies singular graphs by considering minimal singular induced subgraphs of small order. These correspond to a number k of linearly dependent rows of the adjacency matrix determining what is termed as a core of the singular graph. For k at most 5, the distinct cores and corresponding minimal configurations (61 in number) are identified. This provides a method of constructing singular graphs from others of smaller order. Furthermore, it is shown that when a graph has a minimaldoi:10.1016/s0012-365x(97)00036-8 fatcat:pvul5knyqvbrbhxj7fnrtzszy4