As a generalization of strongly regular graphs, van Dam and Omidi [8] introduced the concept of strongly walk-regular graphs. A graph is called strongly ℓ-walk-regular if the number of walks of length ℓ from a vertex to another vertex depends only on whether the two vertices are adjacent, not adjacent, or identical. They proved that this class of graphs falls into several subclasses including connected regular graphs with four eigenvalues, which are called genuine strongly ℓ-walk-regular. In this paper, we prove that the least eigenvalue of a connected genuine strongly 3-walk-regular graph is no more than −2 and characterize all graphs reaching this upper bound.

中文翻译：

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在 true 强 3 游走正则图的最小特征值上


作为强正则图的推广，van Dam 和 Omidi [8] 引入了强游走正则图的概念。如果从一个顶点到另一个顶点的长度为 l 的游走次数仅取决于两个顶点是相邻、不相邻还是相同，则称为强 l-walk-regular。他们证明这类图分为几个子类，包括具有四个特征值的连通正则图，这些特征值称为真正的强 l-walk-regular。在本文中，我们证明了连通的真实强 3 游正则图的最小特征值不超过 -2，并表征了所有达到该上限的图。