Some families of integral graphs

Ligong Wang*, Hajo Broersma, Cornelis Hoede, Xueliang Li, Georg Still

*Corresponding author for this work

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A graph is called integral if all its eigenvalues (of the adjacency matrix) are integers. In this paper, the graphs $K_{1,r}\cdot K_n,\; r^*K_n,\; K_{1,r} \cdot K_{m,n},\; r^*K_{m,n}$ and the tree $K_{1,s}\cdot T(q,r,m,t)$ are defined. We determine the characteristic polynomials of these graphs and also obtain sufficient and necessary conditions for these graphs to be integral. Some sufficient conditions are found by using the number theory and computer search. All these classes are infinite. Some new results which treat interrelations between integral trees of various diameters are also found. The discovery of these integral graphs is a new contribution to the search of such graphs.
Original languageEnglish
Pages (from-to)6383-6391
Number of pages9
JournalDiscrete mathematics
Issue number24
Publication statusPublished - Dec 2008


  • Spectrum
  • Integral tree
  • Integral graph
  • General Pell’s equation

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