The spectral determination of the connected multicone graphs

The main goal of the paper is to answer an unsolved problem. A multicone graph is defined to be the join of a clique and a regular graph, and a wheel as the join of a vertex and a cycle. In this study, we present new classes of multicone graphs that are natural generalizations of wheel graphs and we show that they are determined by their adjacency spectra as well as their Laplacian spectra. We also show that the complements of some of these graphs are determined by their adjacency spectra. In addition, we give a necessary and sufficient condition for perfect graphs cospectral with some of the graphs investigated in the paper. Finally, we conclude with two problems for further study.