Per-spectral characterizations of some edge-deleted subgraphs of a complete graph

Let denote the set of all graphs obtained from by removing five or fewer edges. Camara and Haemers proved that graphs in are uniquely determined by their adjacency spectra with the exception for graphs and . In this paper, we show that all graphs in are uniquely determined by their permanental spectra. We further extend our findings by investigating when a complete graph with a few edges removed is uniquely determined by its permanental spectrum. More precisely, we prove that if induces a star, or a matching, or a disjoint union of a matching and a path , then is uniquely determined by its permanental spectrum.