Integral trees with diameter four

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• 作者：Ligong Wang ; ^a ; ^{lgwangmath@163.com" class="auth_mail" title="E-mail the corresponding author} ; ^{lgwang@nwpu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Qi Wang^b ; ^{wangqi@nwpu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Bofeng Huo^c ; ^{bofenghuo@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Integral tree ; Adjacency matrix ; Diophantine equation ; Graph spectrum
• 刊名：Applied Mathematics and Computation
• 出版年：2016
• 出版时间：5 May 2016
• 年：2016
• 卷：282
• 期：Complete
• 页码：53-64
• 全文大小：512 K

文摘

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;m_i)=S(a₁+a₂+⋯+a_s;m₁,m₂,…,m_s)$S(r;m_i)=S(a_1+a_2+\cdots +a_s;m_1,m_2,\dots ,m_s)$ of diameter 4 with 11dcfcf1" title="Click to view the MathML source">s=3,4,5,6$s=3,4,5,6$. 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^′;m_i)=S(b₁+⋯+b_s;m₁,$S(r^{\prime };m_i)=S(b_1+\cdots +b_s;m_1,$…,$\dots ,$m_s  ) of diameter 4 from given integral trees S(r;m_i)=S(a₁+⋯+a_s;m₁,$S(r;m_i)=S(a_1+\cdots +a_s;m_1,$…,$\dots ,$m_s  ) of diameter 4 are given. Further, using these conditions we construct infinitely many new classes of integral trees S(r^′;m_i)=S(b₁+⋯+b_s;m₁,$S(r^{\prime };m_i)=S(b_1+\cdots +b_s;m_1,$…,$\dots ,$m_s  ) of diameter 4 with 11dcfcf1" title="Click to view the MathML source">s=3,4,5,6$s=3,4,5,6$. Finally, we propose two basic open problems about integral trees of diameter 4 for further study.