The Number of All <em>m</em>-Independent Sets of <em>K</em> Disjoint 3-Regular Graphs with a Given Vertex
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Let K be a positive integer and 1 ≤ K ≤ k, where k is a given positive integer. Consider a 3-regular graph with 8 vertices and denoted by Q3. In addition, KQ3 is composed of K disconnected 3-regular graphs Q3. In this paper, we first consider the number of all m-independent sets of kQ3 with 1 ≤ m ≤ 4k, denoted by Tm(kQ3). Then, we consider the number of all m-independent sets of KQ3 with 1 ≤ K ≤ k-1, denoted by Sm(KQ3).