  Let $P_n$ and $C_n$ denote the path and cycle on $n$ vertices respectively.
The dumbbell graph, denoted by $D_{p,k,q}$, is the graph obtained from two
cycles $C_p$, $C_q$ and a path $P_{k+2}$ by identifying each pendant vertex of
$P_{k+2}$ with a vertex of a cycle respectively. The theta graph, denoted by
$\Theta_{r,s,t}$, is the graph formed by joining two given vertices via three
disjoint paths $P_{r}$, $P_{s}$ and $P_{t}$ respectively. In this paper, we
prove that all dumbbell graphs as well as theta graphs are determined by their
Laplacian spectra.
