图的$k$ -独立数的惯性和比率型界及其关系
On inertia and ratio type bounds for the $k$-independence number of a graph and their relationship
摘要：
对于$k\ge 1$，图的$k$独立数$α_k$是相互距离大于$k$的顶点的最大数目. Cvetković和Hoffman分别给出了图的（1-）独立数$α(=α_1)$的惯性界和比率界，并推广到了$k$的每一个值.我们发现，对于具有足够的规律性的图，多项式参与这样的推广是密切相关的，并给出精确的值为$α_k$，显示了一个新的惯性和比率类型的界限之间的关系。此外，我们还研究了行走-正则图中相互距离最大的顶点集的极值的存在性及其性质。最后，我们利用预距离多项式得到了部分行走正则图的新的锐惯性和比率型界。
英文摘要：
For $k\ge 1$, the $k$-independence number $α_k$ of a graph is the maximum number of vertices that are mutually at distance greater than $k$. The well-known inertia and ratio bounds for the (1-)independence number $α(=α_1)$ of a graph, due to Cvetković and Hoffman, respectively, were generalized recently for every value of $k$. We show that, for graphs with enough regularity, the polynomials involved in such generalizations are closely related and give exact values for $α_k$, showing a new relationship between the inertia and ratio type bounds. Additionally, we investigate the existence and properties of the extremal case of sets of vertices that are mutually at maximum distance for walk-regular graphs. Finally, we obtain new sharp inertia and ratio type bounds for partially walk-regular graphs by using the predistance polynomials.
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