### QLS-Integrality of Complete r-Partite Graphs

#### Abstract

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 multipartite graphs Kb1·p1,b2·p2,...,bs·ps from given QLS-integral r-partite graphs Ka1·p1,a2·p2,...,as·ps are given. Using these conditions we construct new infinite classes of such graphs for s = 4, 5, 6, which affirmatively answers to questions given in [L. Wang, G. Zhao, K.Li: Seidel Integral Complete r-Partite Graphs. Graphs Comb. (2013), DOI:10.1007/s00373-1276-6. and G. Zhao, L.Wang, K.Li: Q-integral complete r-partite graphs. Linear Algebra Appl. 438,1067-1077 (2013)]. Finally, we propose open problems for further study.

conditions for a construction of infinite families of QLS-integral complete multipartite graphs Kb1·p1,b2·p2,...,bs·ps from given QLS-integral r-partite graphs Ka1·p1,a2·p2,...,as·ps are given. Using these conditions we construct new infinite classes of such graphs for s = 4, 5, 6, which affirmatively answers to questions given in [L. Wang, G. Zhao, K.Li: Seidel Integral Complete r-Partite Graphs. Graphs Comb. (2013), DOI:10.1007/s00373-1276-6. and G. Zhao, L.Wang, K.Li: Q-integral complete r-partite graphs. Linear Algebra Appl. 438,1067-1077 (2013)]. Finally, we propose open problems for further study.

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