Abstract:  Let $H(p,tK_{1,m})$ be a connected unicyclic graph with $p+mt$ vertices obtained from $C_{p}$ by attaching the center of star $K_{1,m}$ to each one of $t$ mutual adjacent vertices of the cycle $C_{p}$, respectively. In this paper, it is proved that the unicyclic graphs $H(p,p K_{1,4})$, $H(p,p K_{1,3})$, $H((p,(p-1) K_{1,3} )$ are determined by their Laplacian spectra, and when $ p $ is even number, the unicyclic graphs $H(p,2 K_{1,3})$, $H(p,(p-2)K_{1,3})$, $H(p,(p-3)K_{1,3})$ are also determined by their Laplacian spectra.

Key words: Laplacian spectrum, adjacency spectrum, unicyclic graph