Edge Switching Transformations of Quantum Graphs

Abstract

Discussed here are the effects of basics graph transformations on the spectra of associated quantum graphs. In particular it is shown that under an edge switch the spectrum of the transformed Schrodinger operator is interlaced with that of the original one. By implication, under edge swap the spectra before and after the transformation, denoted by E-n(n=1)(infinity) and (E) over bar (n)(n=1)(infinity) correspondingly, are level-2 interlaced, so that En-2 <= (E) over bar (n) <= En+2. The proofs are guided by considerations of the quantum graphs’ discrete analogs.