垂悬路径与积分广义日图

中文摘要 AI 辅助

如果一个图的邻接矩阵的谱完全由整数组成，则该图称为积分图。我们证明，每个具有至少三条边的垂悬路径的简单图都有一个特征值在 $(1,2\cos(\pi/9)]$ 中，以及一个特征值在 $[-2\cos(\pi/9),-1)$ 中，因此不是积分图。这解决了 Braga、Del-Vecchio 和 Rodrigues（2021）关于积分广义日图的猜想。论证是矩阵理论的：将一条终端路径附加到任意实对称矩阵的三个新坐标上会产生相同的谱障碍，并且上述正区间在这种一般性下是最优的。然后，我们反驳了同一作者的第二个猜想，该猜想断言除圈以外的积分广义日图的圈的长度能被四整除。从六边形出发，在其六个顶点中的五个上分别附加 $6,6,12,6,6$ 个垂悬顶点得到的图是积分图，且有 $42$ 个顶点。我们证明它是受 Pell 方程 $x^{2}-2k^{2}=-7$ 支配的无限族的最小成员，并且我们以闭式形式计算了任意偶圈上类似图的特征多项式。该族内的积分性迫使圈为正方形或六边形，而正方形情形产生了由 Pell 方程 $k^{2}-2c^{2}=1$ 支配的第二个无限族。

英文摘要

A graph is integral if the spectrum of its adjacency matrix consists entirely of integers. We prove that every simple graph having a pendant path with at least three edges has an eigenvalue in $(1,2\cos(π/9)]$ and one in $[-2\cos(π/9),-1)$, and hence is not integral. This settles a conjecture of Braga, Del-Vecchio and Rodrigues (2021) on integral generalized sun graphs. The argument is matrix-theoretic: adjoining a terminal path on three new coordinates to an arbitrary real symmetric matrix produces the same spectral obstruction, and the positive interval above is optimal in this generality. We then disprove a second conjecture of the same authors, which asserts that the cycle of an integral generalized sun graph other than a cycle has length divisible by four. The graph obtained from a hexagon by attaching $6,6,12,6,6$ pendant vertices to five of its six vertices is integral and has $42$ vertices. We show that it is the smallest member of an infinite family governed by the Pell equation $x^{2}-2k^{2}=-7$, and we compute in closed form the characteristic polynomial of the analogous graphs on an arbitrary even cycle. Integrality within this family forces the cycle to be a square or a hexagon, and the square case yields a second infinite family governed by the Pell equation $k^{2}-2c^{2}=1$.

发表机构

• Instituto de Matemática e Estatística, Universidade Federal do Rio Grande do Sul（南里奥格兰德联邦大学数学与统计学院）

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