Digraphs with Hermitian spectral radius below 2 and their cospectrality with paths
Digraphs with Hermitian spectral radius below 2 and their cospectrality with paths
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DOI：
10.1016/j.disc.2017.01.018
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发表时间：
2017-11
期刊：
影响因子：
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通讯作者：
Krystal Guo;B. Mohar
中科院分区：
文献类型：
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作者：
Krystal Guo;B. Mohar
众所周知，路径是由邻接矩阵的谱决定的。对于有向图，其底层图是树的每个有向图都与其底层图关于埃尔米特邻接矩阵（H-共谱）共谱。因此，其底层图与 P n 同构的每个（简单）有向图都是 P n 的 H 共谱。有趣的是，还有其他的。本文发现当n为奇数时与路径图P n 具有H共谱且其底层图非同构的有向图，并发现当n为偶数时此类图不存在。为了证明这个结果，确定了厄米特谱半径小于2的所有有向图。
It is well-known that the paths are determined by the spectrum of the adjacency matrix. For digraphs, every digraph whose underlying graph is a tree is cospectral to its underlying graph with respect to the Hermitian adjacency matrix (H-cospectral). Thus every (simple) digraph whose underlying graph is isomorphic to P n is H-cospectral to P n. Interestingly, there are others. This paper finds digraphs that are H-cospectral with the path graph P n and whose underlying graphs are nonisomorphic, when n is odd, and finds that such graphs do not exist when n is even. In order to prove this result, all digraphs whose Hermitian spectral radius is smaller than 2 are determined.