# THE SPECTRAL DETERMINATIONS OF THE JOIN OF TWO FRIENDSHIP GRAPHS

• Abdian, Ali Zeydi (Department of the Mathematical science, Lorestan University College of Science) ;
• Moez, Amirhossein Morovati (Department of Mathematics, Payame Noor University)
• Accepted : 2019.02.25
• Published : 2019.03.25
• 268 8

#### Abstract

The main aim of this study is to characterize new classes of multicone graphs which are determined by their adjacency spectra, their Laplacian spectra, their complement with respect to signless Laplacian spectra and their complement with respect to their adjacency spectra. A multicone graph is defined to be the join of a clique and a regular graph. If n is a positive integer, a friendship graph $F_n$ consists of n edge-disjoint triangles that all of them meet in one vertex. It is proved that any connected graph cospectral to a multicone graph $F_n{\nabla}F_n=K_2{\nabla}nK_2{\nabla}nK_2$ is determined by its adjacency spectra as well as its Laplacian spectra. In addition, we show that if $n{\neq}2$, the complement of these graphs are determined by their adjacency spectra. At the end of the paper, it is proved that multicone graphs $F_n{\nabla}F_n=K_2{\nabla}nK_2{\nabla}nK_2$ are determined by their signless Laplacian spectra and also we prove that any graph cospectral to one of multicone graphs $F_n{\nabla}F_n$ is perfect.

#### Keywords

Adjacency spectrum;Laplacian spectrum;Multicone graph;DS graph;Friendship graph;Signless Laplacian spectrum

#### File

Figure 1. Friendship graphs F1, F2, F3 and Fn.

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