On triangle-free distance-regular graphs with an eigenvalue multiplicity equal to the valency
On triangle-free distance-regular graphs with an eigenvalue multiplicity equal to the valency
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DOI：
10.1016/j.ejc.2007.06.010
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发表时间：
2008-07
期刊：
影响因子：
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通讯作者：
K. Coolsaet;Aleksandar Jurisic;J. Koolen
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文献类型：
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作者：
K. Coolsaet;Aleksandar Jurisic;J. Koolen
设Γ是直径d≥3，度k≥3，交数a2 ≥ 0的无三角距离正则图.假设Γ有一个重数为k的特征值。证明了当d=3或d≥4且a4=0时，Γ是Nomura意义下的1-齐次的.在后一种情况下，我们证明了Γ是强正则图的对极覆盖，这意味着它的直径为4或5。对于d=5，已知以下无限族的可行相交数组：。当μ=1时，相交阵列由十二面体唯一地实现。对于μ π ι 1，我们证明了不存在具有这个交数组的距离正则图。
Let Γ be a triangle-free distance-regular graph with diameter d≥3, valency k≥3 and intersection number a2≠0. Assume Γ has an eigenvalue with multiplicity k. We show that Γ is 1-homogeneous in the sense of Nomura when d=3 or when d≥4 and a4=0. In the latter case we prove that Γ is an antipodal cover of a strongly regular graph, which means that it has diameter 4 or 5. For d=5 the following infinite family of feasible intersection arrays: is known. For μ=1 the intersection array is uniquely realized by the dodecahedron. For μ≠1 we show that there are no distance-regular graphs with this intersection array.