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Lert $G$ be a simple graph. A $(k,\tau)$-regular set $S$  is a subset of the vertices of a graph $G$, inducing a $k$-regular subgraph such that every vertex not in the subset has $\tau$ neighbors in it. We consider the signless Laplacian spectra of graphs with $(k,\tau)$-regular sets. We give several lower bounds on the index of such graphs as well as some upper bounds on the sum of the squares of the entries of the principal eigenvector corresponding to the vertices in $S$.

graphs, signless Laplacian spectra