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.
Categories: math.CO