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A graph is singular if the zero eigenvalue is in the spectrum of its \(0-1\) adjacency matrix \(A\). If an eigenvector belonging to the zero eigenspace of \(A\) has no zero entries, then the singular graph is said to be a core graph. A \((\kappa, \tau)\)-regular set is a subset of the vertices inducing a \(\kappa\)-regular subgraph such that every vertex not in the subset has \(\tau\) neighbors in it. We consider the case when \(\kappa = \tau\), which relates to the eigenvalue zero under certain conditions. We show that if a regular graph has a \((\kappa, \kappa)\)-regular set, then it is a core graph. By considering the walk matrix, we develop an algorithm to extract \((\kappa, \kappa)\)-regular sets and formulate a necessary and sufficient condition for a graph to be Hamiltonian.