QLS-Integrality of Complete $r$-Partite Graphs

Milan Pokorn\'y

A graph $G$ is called A-integral (L-integral, Q-integral, S-integral) if the spectrum of its adjacency (Laplacian, signless Laplacian, Seidel) matrix consists entirely of integers. In this paper we study connections between the Q- (L,S,A) integral complete multipartite graphs. Moreover, new sufficient conditions for a construction of infinite families of QLS-integral complete $r''$-partite graphs $K_{p_1,p_2,\ldots,p_{r''}}=K_{b_1\cdot p_1,b_2\cdot p_2,\ldots,b_s\cdot p_s}$ from given QLS-integral $r'$-partite graphs $K_{p_1,p_2,\ldots,p_{r'}}=K_{a_1\cdot p_1,a_2\cdot p_2,\ldots,a_s\cdot p_s}$ are given. Using these conditions new infinite classes of such graphs for $s=4,5,6$ are constructed, which affirmatively answers to questions proposed by Wang, Zhao and Li in [10,14]. Finally, we propose open problems for further study.