In this study we investigate the spectra of the family of connected multicone graphs. A multicone graph is defined to be the join of a clique and a regular graph. Let $ r $, $ t $ and $ s $ be natural numbers, and let $ K_r $ denote a complete graph on $ r $ vertices. It is proved that connected multicone graphs $ K_r\bigtriangledown sK_t $, a natural generalization of friendship graphs, are determined by their adjacency spectra as well as their Laplacian spectra. Also, we show that the complement of multicone graphs $ K_r\bigtriangledown sK_t $ are determined by their adjacency spectra, where $ s\neq 2 $.