Given a compact quantum graph, length swapping is the operation where two edge lengths are swapped. Swapping changes neither the total length of the graph nor the connectivity. Switching is the same operation as in combinatorial graph theory, and the switched edges retain their lengths. Switching conserves the total length and the degrees of the vertices but changes the connectivity. Both operations alter the graph spectrum if the lengths are not commensurate (and in the absence of symmetries).

The lecture will deal with the effect of edge swapping and switching on the graph spectrum. The main result is that the spectra of the original and switched/swapped graphs interlace: denote by \(k^2_n\) and \(\tilde{k}^2_n\) the \(n^{\text{th}}\) spectral points in the ordered spectra; then \( k_{n-2} ≤ \tilde{k}_n ≤ k_{n+2} \) for all \(n > 2\). In other words, if \(N(k)\) and \(\tilde N (k)\) denote the spectral counting functions, then \( |N(k) − \tilde N (k)| ≤ 2\).

Extensions and application of the swapping and switching operations in various spectral problems will be discussed.