Integral trees with diameter four

Ligong Wang, Qi Wang and Bofeng Huo

Applied Mathematics and Computation, 2016, vol. 282, issue C, 53-64

Abstract: A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. In this paper, we investigate integral trees S(r;mi)=S(a1+a2+⋯+as;m1,m2,…,ms) of diameter 4 with s=3,4,5,6. Such integral trees are found by using a computer search or solving the Diophantine equations. New sufficient conditions for a construction of infinite families of integral trees S(r′;mi)=S(b1+⋯+bs;m1,…,ms) of diameter 4 from given integral trees S(r;mi)=S(a1+⋯+as;m1,…,ms) of diameter 4 are given. Further, using these conditions we construct infinitely many new classes of integral trees S(r′;mi)=S(b1+⋯+bs;m1,…,ms) of diameter 4 with s=3,4,5,6. Finally, we propose two basic open problems about integral trees of diameter 4 for further study.

Keywords: Integral tree; Adjacency matrix; Diophantine equation; Graph spectrum (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:282:y:2016:i:c:p:53-64

DOI: 10.1016/j.amc.2016.02.002

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