A geometric construction of isospectral magnetic graphs

Abstract

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.

Figures (10)

• Figure 1: The construction of pairs of isospectral graphs: Top row from left to right: the building block $\bm G$ (a so-called kite graph), the frame members $\bm F_1=\bm G$, $\bm F_2$ and $\bm F_3$. Bottom row: The two graphs $\bm F_{A,V_1}$ and $\bm F_{B,V_1}$ are isospectral but not isomorphic. Note that this is the smallest possible choice of non-trivial partitions, namely the two different $2$-partitions of $A=\lMset 1,3\rMset$ (left) and $B=\lMset 2,2\rMset$ (right) ($4=1+3=2+2$). Note that the kite graph can also carry a magnetic potential; here given by three parameters, see Example \ref{['ex:kite']}.
• Figure 2: The frame $(\bm F^\theta_a)_{a \in \mathbb{N}}$ given by the frame members $\bm F^\theta_a$ for $a\in \mathbb{N}$. Each graph $\bm F^\theta_a$ has a so-called distinguished bottom vertex (outlined); it will be used in the next step for the contracted frame union.
• Figure 3: The contracted frame unions $\bm F^\theta_{A,v_1}$ and $\bm F^\theta_{B,v_1}$ for the two different $2$-partitions $A=\lMset 1,3\rMset$ and $B=\lMset 2,2\rMset$ of $4$ are isospectral, but not isomorphic for each value $\theta \in \mathbb{R}/2\pi\mathbb{Z}$ of the magnetic potential.
• Figure 4: The three types of eigenfunctions (represented by dotted vertical lines with arrow pointing in a virtual third dimension); Dirichlet vertices are marked outlined with a bigger circle: Top: The eigenfunctions of $\bm G^\theta$ are copied symmetrically onto each copy of $\bm G^\theta$ in $\bm F^\theta_{A,v_1}$. Hence, the three eigenfunctions (in the picture, it is the constant one) become three eigenfunctions on $\bm F^\theta_{A,v_1}$. Middle: each eigenfunction of $(\bm G^\theta)^+_{v_1}$ becomes a symmetric copy on each member $(\bm F_a^\theta)^+_{v_1}$ of $\bm F^\theta_{A,v_1}$ for $a \in A=\lMset 3,1\rMset$. To make them orthogonal to the symmetric ones, we can choose only $s-1=1$ one here for each of the two eigenfunctions of $(\bm G^\theta)^+_{v_1}$, hence $2(s-1)=2$ eigenvalues of $\bm F^\theta_{A,v_1}$ are captured. Bottom: we consider the eigenfunctions of $(\bm G^\theta)^+_{V_0}$ (here only one) onto each copy of $(\bm G^\theta)^+_{V_0}$ in $(\bm F_a^\theta)^+_{v_1}$ for $a \in \lMset 3,1\rMset$. There are here $r-s=(3-1)+(1-1)=2$ such eigenfunctions orthogonal to the previous ones, supported here only on $(\bm F_3^\theta)^+_{v_1}$.
• Figure 5: The graphs $\bm F_{A_q,v_1}^\theta$ defined by the $4$-partitions $A_1=\lMset1,1,1,5\rMset$, $A_2=\lMset1,1,2,4\rMset$, $A_3=\lMset1,1,3,3\rMset$, $A_4=\lMset1,2,2,3\rMset$ and $A_5=\lMset2,2,2,2\rMset$ of $r=8$. All graphs have $r+s+1=13$ vertices ($s=4$) and $3r=24$ edges. There are five different $4$-partitions of $8$ (all listed above).

Theorems & Definitions (42)

• Definition 2.1: Contracting vertices and shrinking number
• Definition 2.2: discrete (normalised) magnetic Laplacian and its spectrum
• Remark 2.3: matrix representation of the Laplacian
• Definition 2.4: isospectral magnetic graphs
• Remark 2.5: multiple edges versus weighted graphs without multiple edges
• Remark 2.6: signed graphs as a special case of magnetic graphs
• Proposition 2.7: magnetic Laplacians and cohomologous magnetic potentials
• proof
• Definition 2.8: (normalised) magnetic Dirichlet Laplacian
• Proposition 2.9