I am wondering the following. If we add a pendant vertex to a graph we are changing the spectrum. Eigenvalue interlacing tells us that the largest e-value of the new graph is at least as big as that of the old, and the smallest is at least as small, but what I'm wondering is whether they can both be equal. I feel somehow that either the largest should be strictly larger or the smallest should be strictly smaller, or both. I'm particularly interested in the bipartite case, because in that case both spectra are symmetric, so if the largest e-value is strictly larger than the smallest must be strictly smaller. Does anyone know about this? My guess could also be wrong, but I don't see how to prove it or find a counterexample.
Thanks, Greg