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

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

Research output: Contribution to journalArticlepeer-review


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
Issue number1
Publication statusPublished - 01-06-2019

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics

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