The Hermitian two-graph and its code

Haemers, Willem H.; Kuijken, Elisabeth

doi:10.1016/s0024-3795(02)00320-8

Linear Algebra and its Applications

2002

DOI: 10.1016/s0024-3795(02)00320-8

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The Hermitian two-graph and its code

Willem H. Haemers

Elisabeth Kuijken

Abstract: DevwudfwWkh wzr0judsk frgh ri wkh Khuplwldq wzr0judsk K+t, lv lqyhvwljdwhg1 E| xvh ri wklv frgh zh vkrz wkdw li t 6 +prg ;, wkhuh h{lvwv qr vwurqjo| uhjxodu judsk zlwk ydohqf| t 2 +t 4,@5 lq wkh vzlwfklqj fodvv fruuhvsrqglqj wr K+t,1 Iru t @ 8 wkh frgh lv zrunhg rxw lq ghwdlo1 Zh jhqhudwh wkh zhljkw hqxphudwru dqg jlyh dq dffrxqw iru wkh frgh zrugv ri vhyhudo zhljkwv1 Khuh lw iroorzv wkdw wkhuh lv d xqltxh uhjxodu judsk zlwk ydohqf| 83 lq wkh vzlwfklqj fodvv1Nh|zrugv= glvfuhwh pdwkhpdwlfv/ qlwh jhrphwu|/ vwurq… Show more

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Cited by 3 publications

References 15 publications

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Ball packings with high chromatic numbers from strongly regular graphs

Chen¹

2017

Discrete Mathematics

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Inspired by Bondarenko's counter-example to Borsuk's conjecture, we notice some strongly regular graphs that provide examples of ball packings whose chromatic numbers are significantly higher than the dimensions. In particular, from generalized quadrangles we obtain unit ball packings in dimension q 3 − q 2 + q with chromatic number q 3 + 1, where q is a prime power. This improves the previous lower bound for the chromatic number of ball packings. The problem and previous worksA ball packing in d-dimensional Euclidean space is a collection of balls with disjoint interiors. The tangency graph of a ball packing takes the balls as vertices and the tangent pairs as edges. The chromatic number of a ball packing is defined as the chromatic number of its tangency graph.The Koebe-Andreev-Thurston disk packing theorem says that every planar graph is the tangency graph of a 2-dimensional ball packing. The following question is asked by Bagchi and Datta in [BD13] as a higher dimensional analogue of the four-colour theorem:Problem. What is the maximum chromatic number χ(d) over all the ball packings in dimension d?The authors gave d + 2 ≤ χ(d) as a lower bound since it is easy to construct d + 2 mutually tangent balls. By ordering the balls by size, the authors also argued that κ(d) + 1 is an upper bound, where κ(d) is the kissing number for dimension d.However, the case of d = 3 has already been investigated by Maehara [Mae07], who proved that 6 ≤ χ(3) ≤ 13. His construction for the lower bound uses a variation of Moser's spindle, which is the tangency graph of an unit disk packing in dimension 2 with chromatic number 4, and the following lemma:Lemma. If there is a unit ball packing in dimension d with chromatic number χ, then there is a ball packing in dimension d + 1 with chromatic number χ + 2.

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Ball packings with high chromatic numbers from strongly regular graphs

Chen¹

2017

Discrete Mathematics

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Using Algebraic Properties of Minimal Idempotents for Exhaustive Computer Generation of Association Schemes

Coolsaet¹,

Degraer²

2008

Electron. J. Combin.

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During the past few years we have obtained several new computer classification results on association schemes and in particular distance regular and strongly regular graphs. Central to our success is the use of two algebraic constraints based on properties of the minimal idempotents $E_{i}$ of these association schemes : the fact that they are positive semidefinite and that they have known rank. Incorporating these constraints into an actual isomorph-free exhaustive generation algorithm turns out to be somewhat complicated in practice. The main problem to be solved is that of numerical inaccuracy: we do not want to discard a potential solution because a value which is close to zero is misinterpreted as being negative (in the first case) or nonzero (in the second). In this paper we give details on how this can be accomplished and also list some new classification results that have been recently obtained using this technique: the uniqueness of the strongly regular $(126,50,13,24)$ graph and some new examples of antipodal distance regular graphs. We give an explicit description of a new antipodal distance regular $3$-cover of $K_{14}$, with vertices that can be represented as ordered triples of collinear points of the Fano plane.

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On Base Partitions and Cover Partitions of Skew Characters

Gutschwager¹

2008

Electron. J. Combin.

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In this paper we give an easy combinatorial description for the base partition ${\cal B}$ of a skew character $[{\cal A}]$, which is the intersection of all partitions $\alpha$ whose corresponding character $[\alpha]$ appears in $[{\cal A}]$. This we use to construct the cover partition ${\cal C}$ for the ordinary outer product as well as for the Schubert product of two characters and for some skew characters, here the cover partition is the union of all partitions whose corresponding character appears in the product or in the skew character. This gives us also the Durfee size for arbitrary Schubert products.

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The Hermitian two-graph and its code

Cited by 3 publications

References 15 publications

Ball packings with high chromatic numbers from strongly regular graphs

Ball packings with high chromatic numbers from strongly regular graphs

Using Algebraic Properties of Minimal Idempotents for Exhaustive Computer Generation of Association Schemes

On Base Partitions and Cover Partitions of Skew Characters

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