Integral complete r-partite graphs

Ligong Wang, Xueliang Li, Cornelis Hoede

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A graph is called integral if all the eigenvalues of its adjacency matrix are integers. In this paper, we give a useful sufficient and necessary condition for complete r-partite graphs to be integral, from which we can construct infinite many new classes of such integral graphs. It is proved that the problem of finding such integral graphs is equivalent to the problem of solving some Diophantine equations. The discovery of these integral complete r-partite graphs is a new contribution to the search of such integral graphs. Finally, we propose several basic open problems for further study.
Original languageEnglish
Pages (from-to)231-241
JournalDiscrete mathematics
Issue number1-3
Publication statusPublished - 2004


  • Diophantine equation
  • Integral graph
  • Graph spectrum
  • Complete r-partite graph

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