Integral trees of arbitrarily large diameters
Péter Csikvári
DOI: 10.1007/s10801-010-0218-8
Abstract
In this paper, we construct trees having only integer eigenvalues with arbitrarily large diameters. In fact, we prove that for every finite set S of positive integers there exists a tree whose positive eigenvalues are exactly the elements of S. If the set S is different from the set {1} then the constructed tree will have diameter 2| S|.
Pages: 371–377
Keywords: keywords trees; eigenvalues
Full Text: PDF
References
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2. Brouwer, A.E.: Small integral trees. Electron. J. Comb. 15 (2008)
3. Brouwer, A.E., Haemers, W.H.: The integral trees with spectral radius
3. Linear Algebra Appl. 429, 2710-2718 (2008)
4. Cao, Z.F.: Integral trees of diameter R (3 \leq R \leq 6). Heilongjiang Daxue Ziran Kexue Xuebao 95(2), 1-3 (1988)
5. Cao, Z.F.: Some new classes of integral trees with diameter 5 or
6. J. Syst. Sci. Math. Sci. 11, 20-26 (1991)
6. Csikvári, P.: Short note on the integrality of some trees.
7. Godsil, C., Royle, G.: Algebraic Graph Theory. Graduate Texts in Mathematics. Springer, Berlin (2001), pp. 186
8. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. A Foundation for Computer Science, 2nd edn. Addison-Wesley, Reading (1994)
9. Harary, F., Schwenk, A.J.: Which graphs have integral spectra? In: Graph and Combinatorics. Lecture Notes in Mathematics, vol. 406, pp. 45-51. Springer, Berlin (1974)
10. Wang, L.G., Li, X.L.: Some new classes of integral trees with diameters 4 and
6. Australas. J. Comb.
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