Spectral characterization of new classes of multicone graphs

Abstract

This paper deals with graphs that are known as multicone graphs. A multicone graph is a graph obtained from  the join of a clique and a regular graph. Let $ w $, $ l $, $ m $ be natural numbers and  $ k$ is a natural number. It is proved that any connected graph cospectral with multicone graph $K_w\bigtriangledown mECP_{l}^{k}$ is determined by its adjacency spectra as well as its Laplacian spectra, where   $ ECP_{l}^{k}={K_{\underbrace {{3^k},\,{3^k},\,...,\,{3^k}}_{l\,times}}}$. Also, we show that   complements of some of these multicone graphs are determined by their adjacency spectra.Moreover,  we prove that any connected graph cospectral with these multicone graphs must be perfect. Finally, we pose two  problems for further researches.