Abstract
Abstract The problem of finding the maximum independent sets (or maximum cliques) of a given graph is fundamental in graph theory and is also one of the most important in terms of the application of graph theory. Let $A(n,d,w)$ be the size of the maximum independent set of $Q_{n}^{(d-1,w)}$, which is the induced subgraph of points of weight $w$ of the $d-1^{th}$-power of $n$-dimensional hypercubes. In order to further understand and study the dependent set of $Q_{n}^{(d-1,w)}$, we explore its clique number and the structure of the maximum clique. This paper obtains the clique number and the structure of the maximum clique of $Q_{n}^{(d-1,w)}$ for $5\leq d\leq 6$. Moreover, the characterizations for $A(n,d,w)=2$ and $3$ are also given.
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