AIMS Mathematics, 2019, 4(5): 1348-1356. doi: 10.3934/math.2019.5.1348

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The spectral determinations of connected multicone graphs KwmCP(n)

1 Department of Mathematics, Payame Noor University, PO BOX 19395-3697, Tehran, Iran;
2 Department of the Mathematical science, Lorestan University, College of Science, Lorestan, Khoramabad, Iran

The main goal of this study is to characterize new classes of multicone graphs which are determined by their spectra. One of important part of algebraic graph theory is devoted to spectral graph theory. Determining whether a graph is determined by its spectra or not is often an important and challenging problem. In [1] it have been shown that the join of a Cocktail-Party graph with an arbitrary complete graph is determined by both its adjacency spectra and its Laplacian spectra. In this work, we aim to generalize these facts. A multicone graph is defined to be the join of a clique and a regular graph. Let w, m and n be natural numbers. In this paper, it is proved that any connected graph cospectral to a multicone graph KwmCP(n) is determined by its adjacency spectra as well as its Laplacian spectra, where $ CP(n)={K_{\underbrace {2,\,.\,.\,.,\,\,2}_{n\,times}}}$ is a Cocktail-Party graph. Moreover, we prove that any graph cospectral to one of these multicone graphs must be perfect. Finally, we pose two conjectures for further research.
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