Abstract
Abstract This paper presents results for some vertex stress related parameters in respect of specific subfamilies of Kneser graphs. Kneser graphs for which diam(KG(n, k)) = 2 and k ≥ 2 are considered. The note establishes the foundation for researching similar results for Kneser graphs for which diam(KG(n, k)) ≥ 3. In addition some important vertex stress related properties are stated. Finally some results for specific bipartite Kneser graphs i.e. BK(n, 1), n ≥ 3 will be presented. In the conclusion some worthy research avenues are proposed.
Highlights
It is assumed that the reader has good working knowledge of set theory
K ≥ 1 be the k-element subsets of the set, Computing Classification System 1998: G.2.2 Mathematics Subject Classification 2010: 05C12, 05C30, 05C69 Key words and phrases: vertex stress, diameter, distance, Kneser graph
A Kneser graph denoted by KG(n, k), n, k ∈ N is a graph with vertex set, V(KG(n, k)) = {vi : vi → Xi}
Summary
It is assumed that the reader has good working knowledge of set theory. For the general notation, notions and important introductory results in set theory, see [3]. Corollary 3 For a Kneser graph KG(n, 2), n ≥ 5 the total induced vertex stress of vi ∈ V(KG(n, 2)) is given by, sKG(n,2)(vi) =. Theorem 6 For a Kneser graph KG(n, k1), k1 ∈ N\{1, 2}, n ≥ 3k1 − 1 the total induced vertex stress of vi ∈ V(KG(n, k1)) is given by sKG(n,k1)(vi) =. Corollary 7 For a Kneser graph KG(n, k1), k1 ∈ N\{1, 2}, n ≥ 3k1 − 1 the total vertex stress is given by. A general result (without further proof) is permitted from the knowledge that all Kneser graphs KG(n, k), n ≥ k are vertex transitive. Theorem 13 All Kneser graphs KG(n, k), n ≥ k are stress regular
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