论文标题
等光谱磁图的几何结构
A geometric construction of isospectral magnetic graphs
论文作者
论文摘要
我们介绍了有限等光谱图的几何结构，该家族由给定长度$ s $的自然数量$ r $的不同分区（汇总数）。同志此处是指具有归一化重量（包括标准权重）的离散磁性laplacian。构造以任意有限的图形$ g $开头，具有归一化的重量和磁性电位，作为一个构建块，我们在第一步中构造了所谓的框架图$（f_a）_ {a \ in \ mathbb {n}} $的家族。框架图$ f_a $是构建的，沿顶点$ v_0 $的子集$ a $ a $ a $副本$ g $。在第二步中，对于任何分区$ a =（a_1，\ dots，a_s），长度$ s $的自然数量$ r $（即，$ r = a_1+\ dots+a_s $），我们构建了一个新的图$ f_a $，现在签约fragames $ f_ {a_1} $ f_ {a_1}，\ dots $ a_s $ a_s $ a_s $ a_s $ a_s $ a_s $ a_s $ an $ V_1 \子集V_0 $。由$ r \ geq 4 $的不同$ s $分区获得的所有图表（对于$ v_0 $和$ v_1 $的任何选择）都是等值和非同态。 特别是，我们获得了越来越多的图形家庭，对于给定的$ r $和$ s $，对于不同类型的磁性拉普拉斯人来说是同一谱系，包括标准的拉普拉斯，无名的标准拉普拉斯，某些签名的拉普拉斯人，以及（未绑定的）Kirchhoff Laplacian的吉尔霍夫Laplacian，吉尔霍夫Laplacian的吉尔略·拉普拉斯（Kirchhoff Laplacian）的吉利群落级别均匀分别为小米。等光谱图的光谱取决于Builder Block $ G $的Laplacian的光谱以及在一组dirichlet条件下的Laplacian的频谱，该范围$ v_0 $ v_0 $和$ v_1 $，由数字$ r $和$ r $和$ s $ s $ r $和$ s $。
We present a geometrical construction of families of finite isospectral graphs labelled by different partitions of a natural number $r$ of given length $s$ (the number of summands). Isospectrality here refers to the discrete magnetic Laplacian with normalised weights (including standard weights). The construction begins with an arbitrary finite graph $G$ with normalised weight and magnetic potential as a building block from which we construct, in a first step, a family of so-called frame graphs $(F_a)_{a \in \mathbb{N}}$. A frame graph $F_a$ is constructed contracting $a$ copies of $G$ along a subset of vertices $V_0$. In a second step, for any partition $A=(a_1,\dots,a_s)$ of length $s$ of a natural number $r$ (i.e., $r=a_1+\dots+a_s$) we construct a new graph $F_A$ contracting now the frames $F_{a_1},\dots,F_{a_s}$ selected by $A$ along a proper subset of vertices $V_1\subset V_0$. All the graphs obtained by different $s$-partitions of $r\geq 4$ (for any choice of $V_0$ and $V_1$) are isospectral and non-isomorphic. In particular, we obtain increasing finite families of graphs which are isospectral for given $r$ and $s$ for different types of magnetic Laplacians including the standard Laplacian, the signless standard Laplacian, certain kinds of signed Laplacians and, also, for the (unbounded) Kirchhoff Laplacian of the underlying equilateral metric graph. The spectrum of the isospectral graphs is determined by the spectrum of the Laplacian of the building block $G$ and the spectrum for the Laplacian with Dirichlet conditions on the set of vertices $V_0$ and $V_1$ with multiplicities determined by the numbers $r$ and $s$ of the partition.
