Gap sets for the spectra of cubic graphs

Communications of the American Mathematical Society Pub Date : 2020-05-11 DOI:10.1090/cams/3

Alicia J. Koll'ar, P. Sarnak

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引用次数: 14

Abstract

We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals ( 2 2 , 3 ) (2 \sqrt {2},3) and [ − 3 , − 2 ) [-3,-2) achieved in cubic Ramanujan graphs and line graphs are maximal. We give constraints on spectra in [ − 3 , 3 ] [-3,3] which are maximally gapped and construct examples which achieve these bounds. These graphs yield new instances of maximally gapped intervals. We also show that every point in [ − 3 , 3 ) [-3,3) can be gapped by planar cubic graphs. Our results show that the study of spectra of cubic, and even planar cubic, graphs is subtle and very rich.

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三次图谱的间隙集

研究了大有限三次图邻接矩阵谱中的间隙。已知在三次拉马努金图和线形图中得到的间隙区间(2,2,3)(2 \sqrt{2}，3)和[-3，-2)[-3，-2)是极大的。我们给出了在[−3,3][-3,3]区间的谱的最大间隙约束，并构造了达到这些边界的例子。这些图产生了最大间隔的新实例。我们还证明了[−3,3)[-3,3)]中的每一个点都可以被平面三次图隔开。我们的研究结果表明，三次光谱，甚至是平面三次光谱图的研究是微妙而丰富的。

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Communications of the American Mathematical Society

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