Some properties of the knödel graph w (K, 2k), k ≥ 4

R. Balakrishnan, P. Paulraja, Wasin So, M. Vinay

Research output: Contribution to journalArticle

Abstract

Knödel graphs have, of late, come to be used as strong competitors for hypercubes in the realms of broadcasting and gossiping in interconnection networks. For an even positive integer n and 1 ≤ Δ ≤ ⌊log2 n⌋, the general Knödel graph WΔ,n is the Δ-regular bipartite graph with bipar-tition sets X = {x0, x1, …, xn −1} and Y = {y0, y1, …, yn −1} such that xj is adjacent n . The edge x to yj, yj+2 1 −1,2yj+2 2 −1, …, yj+2Δ−1−1, with2 suffixes being taken modulo2j yj+2i −1 at xj and the edge yj xj−(2i −1) at yj are called edges of dimension i at the stars centered at xj and yj respectively. In this paper, we concentrate on the Knödel graph Wk = Wk,2k with k ≥ 4. We show that for k ≥ 4, any automorphism of Wk fixes the set of 0-dimensional edges of Wk . We determine the automorphism group Aut(Wk) of Wk and show that it is isomorphic to the dihedral group D2k−1. In addition, we determine the spectrum of Wk and prove that it is never integral. As a by-product of our results, we obtain three new proofs showing that, for k ≥ 4, Wk is not isomorphic to the hyper-cube Hk of dimension k, and a new proof for the result that Wk is not edge-transitive.

Original languageEnglish
Pages (from-to)17-32
Number of pages16
JournalAustralasian Journal of Combinatorics
Volume74
Issue number1
Publication statusPublished - 01-06-2019
Externally publishedYes

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All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics

Cite this

Balakrishnan, R., Paulraja, P., So, W., & Vinay, M. (2019). Some properties of the knödel graph w (K, 2k), k ≥ 4. Australasian Journal of Combinatorics, 74(1), 17-32.