论文标题
在广义伯特树的主要范围内调用猜想
Disproof of a conjecture on the main spectrum of generalized Bethe trees
论文作者
论文摘要
如果全部矢量与其相关的特征空间不正交，则据说图的邻接矩阵的特征值被认为是主要的。具有$ k $级别的广义伯特树是一棵根树，在同一水平上的顶点具有相同的程度。 França和Brondani [在广义伯特树的主要光谱上，线性代数应用，628（2021）56-71]最近猜想，任何具有$ k $ lacte的伯特树都有$ k $ k $ main eigenvalues，只要$ k $均匀。我们通过为整数$ k \ ge 6 $构建一个反示例家族来反驳猜想。
An eigenvalue of the adjacency matrix of a graph is said to be main if the all-ones vector is not orthogonal to its associated eigenspace. A generalized Bethe tree with $k$ levels is a rooted tree in which vertices at the same level have the same degree. França and Brondani [On the main spectrum of generalized Bethe trees, Linear Algebra Appl., 628 (2021) 56-71] recently conjectured that any generalized Bethe tree with $k$ levels has exactly $k$ main eigenvalues whenever $k$ is even. We disprove the conjecture by constructing a family of counterexamples for even integers $k\ge 6$.
