### 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 |
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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 |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics

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## Cite this

^{k}), k ≥ 4.

*Australasian Journal of Combinatorics*,

*74*(1), 17-32.