Generating isospectral but not isomorphic quantum graphs

TL;DR

This work tackles the longstanding question of isospectral but non-isomorphic quantum graphs by systematically enumerating isospectral sets of equilateral graphs with Neumann-type boundary conditions using computer algebra. Central to the approach is the Titchmarsh-Weyl M-function, which enables combinatorial generation of large isospectral families through attaching graphs at vertices with identical M-functions, yielding generating sets and infinite constructions (e.g., P, Q, R sets, chain-of-loops, pumpkin graphs). The authors extend the scope to Dirichlet and δ-type boundary conditions, analyze the zero-eigenvalue situation, and provide extensive tests, trees, and even almost-isospectral scenarios, all complemented by open-source software for reproducibility. Overall, the paper delivers both a catalog of isospectral graphs within a restricted class and a robust framework for constructing and probing isospectrality across a broad spectrum of boundary conditions and graph families. This advances understanding of spectral graph theory and provides practical tools for exploring isospectrality in quantum graphs and related systems.

Abstract

Quantum graphs are defined by having a Laplacian defined on the edges of a metric graph with boundary conditions on each vertex such that the resulting operator, $\mathbf{L}$, is self-adjoint. We use Neumann boundary conditions although we do a slight excursion into graphs with Dirichlet and $δ$-type boundary condititons towards the end of the paper. The spectrum of $\mathbf{L}$ does not determine the graph uniquely, that is, there exist non-isomorphic graphs with the same spectra. There are few known examples of pairs of non-isomorphic but isospectral quantum graphs. In this paper we start to correctify this situation by finding hundreds of isospectral sets, using computer algebra. We have found all sets of isospectral but non-isomorphic equilateral connected quantum graphs with at most nine vertices. This includes thirteen isospectral triplets and one isospectral set of four. One of the isospectral triplets involves a loop where we could prove isospectrality. We also present several different combinatorial methods to generate arbitrarily large sets of isospectral graphs, including infinite graphs in different dimensions. As part of this we have found a method to determine if two vertices have the same Titchmarsh-Weyl $M$-function. We give combinatorial methods to generate sets of graphs with arbitrarily large number of vertices with the same $M$-function. We also find several sets of graphs that are isospectral under more general, permutation invariant, boundary conditions. This necessitates a study of eigenvalue zero where we prove several results. We discuss the possibilities that our program is incorrect, present our tests and open source it for inspection at http://github.com/meapistol/Spectra-of-graphs

Figures (46)

• Figure 1: All isospectral pairs with at most seven vertices. The edge length is one for all graphs. a) The one isospectral pair with six vertices. b) - f) The five isospectral pairs with seven vertices. The pairs in e) have one extra edge compared with those in d). In all cases one member of the pair has a terminal vertex and the other not. All graphs in b) - e) are subgraphs of a member of f). Isospectral graphs sometimes have vertices with the same $M$-functions. Such vertices are red. $M$-functions are described in the main text.
• Figure 2: a)-c) The three isospectral triplets with eight vertices. Two vertices among the graphs in b) have the same $M$-function, red. The red vertices in c) have the same $M$-function say $M_1$, and so do the yellow ones, say $M_2$. $M_1 \neq M_2$. d) The single isospectral set of four with nine vertices. Also here we have two sets of vertices having the same $M$-function if they have the same colour (except black). There are graphs with eight vertices belonging to isospectral sets that are not a subgraph of any member of a). The graphs have been checked such that there are no overlapping edges.
• Figure 3: All isospectral pairs of trees with at most ten vertices. a) The isospectral pair with nine vertices. b), c) The two isospectral pairs having ten vertices. These graphs have no two vertices with the same $M$-function within each pair.
• Figure 4: Three very simple isospectral graphs. a) A loop graph having length eight. b) The first isospectral partner of the loop graph when $L_1=L_3=2$ and $L_2=4$ such that the total length of the graph is 8. c) The second isospectral partner of the loop.
• Figure 5: Three simple isospectral pairs where a) is isospectral with b), c) is isospectral with d) and e) is isospectral with f). The length of the edges connecting any two vertices are indicated. Vertices with the same colour have the same $M$-function within each isospectral pair. The isospectral pair in e-f) does not have any hot vertices seen as equilateral graphs with 12 vertices.

Theorems & Definitions (16)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• Theorem 5
• proof
• Theorem 6