Electronic International Standard Serial Number (EISSN)
1664-235X
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 GG 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 (FFa)a∈N . A frame graph FFa is constructed contracting a copies of G along a subset of vertices V0 . In a second step, for any partition A=(a1,…,as) of length s of a natural number r (i.e., r=a1+⋯+as ) we construct a new graph FFA contracting now the frames FFa1,…,FFas selected by A along a proper subset of vertices V1⊂V0 . All the graphs obtained by different s-partitions of r≥4 (for any choice of V0 and V1 ) 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 V0 and V1 with multiplicities determined by the numbers r and s of the partition.
Classification
subjects
Mathematics
keywords
isospectrality; discrete magnetic laplacian; spectral gaps