Abstract
Abstract We consider two types of joins of graphs G 1 {G}_{1} and G 2 {G}_{2} , G 1 ⊻ G 2 {G}_{1}\hspace{0.33em}⊻\hspace{0.33em}{G}_{2} – the neighbors splitting join and G 1 ∨ = G 2 {G}_{1}\mathop{\vee }\limits_{=}{G}_{2} – the nonneighbors splitting join, and compute the adjacency characteristic polynomial, the Laplacian characteristic polynomial, and the signless Laplacian characteristic polynomial of these joins. When G 1 {G}_{1} and G 2 {G}_{2} are regular, we compute the adjacency spectrum, the Laplacian spectrum, the signless Laplacian spectrum of G 1 ∨ = G 2 {G}_{1}\mathop{\vee }\limits_{=}{G}_{2} , and the normalized Laplacian spectrum of G 1 ⊻ G 2 {G}_{1}\hspace{0.33em}⊻\hspace{0.33em}{G}_{2} and G 1 ∨ = G 2 {G}_{1}\mathop{\vee }\limits_{=}{G}_{2} . We use these results to construct nonregular, nonisomorphic graphs that are cospectral with respect to the four matrices: adjacency, Laplacian, signless Laplacian and normalized Laplacian.
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