### 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 ≤ ⌊log_{2} n⌋, the general Knödel graph W_{Δ,n} is the Δ-regular bipartite graph with bipar-tition sets X = {x_{0}, x_{1}, …, x^{n} _{−1}} and Y = {y_{0}, y_{1}, …, y^{n} _{−1}} such that x_{j} is adjacent n . The edge x to y_{j}, y_{j+2} ^{1} _{−1},^{2}y_{j+2} ^{2} _{−1}, …, y_{j+2Δ−1−1}, with^{2} suffixes being taken modulo_{2j} y_{j+2i −1} at x_{j} and the edge y_{j} x_{j−(2i −1)} at y_{j} are called edges of dimension i at the stars centered at x_{j} and y_{j} respectively. In this paper, we concentrate on the Knödel graph W_{k} = W_{k,2k} with k ≥ 4. We show that for k ≥ 4, any automorphism of W_{k} fixes the set of 0-dimensional edges of W_{k} . We determine the automorphism group Aut(W_{k}) of W_{k} and show that it is isomorphic to the dihedral group D_{2k−1}. In addition, we determine the spectrum of W_{k} and prove that it is never integral. As a by-product of our results, we obtain three new proofs showing that, for k ≥ 4, W_{k} is not isomorphic to the hyper-cube H_{k} of dimension k, and a new proof for the result that W_{k} is not edge-transitive.

Original language | English |
---|---|

Pages (from-to) | 17-32 |

Number of pages | 16 |

Journal | Australasian Journal of Combinatorics |

Volume | 74 |

Issue number | 1 |

Publication status | Published - 01-06-2019 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics

### Cite this

^{k}), k ≥ 4.

*Australasian Journal of Combinatorics*,

*74*(1), 17-32.

}

^{k}), k ≥ 4',

*Australasian Journal of Combinatorics*, vol. 74, no. 1, pp. 17-32.

**Some properties of the knödel graph w (K, 2 ^{k}), k ≥ 4.** / Balakrishnan, R.; Paulraja, P.; So, Wasin; Vinay, M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Some properties of the knödel graph w (K, 2k), k ≥ 4

AU - Balakrishnan, R.

AU - Paulraja, P.

AU - So, Wasin

AU - Vinay, M.

PY - 2019/6/1

Y1 - 2019/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85068729305&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068729305&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85068729305

VL - 74

SP - 17

EP - 32

JO - Australasian Journal of Combinatorics

JF - Australasian Journal of Combinatorics

SN - 1034-4942

IS - 1

ER -

^{k}), k ≥ 4. Australasian Journal of Combinatorics. 2019 Jun 1;74(1):17-32.