Spectral characterizations of almost complete graphs

TL;DR

This paper explores when almost complete graphs are uniquely identified by their spectra, showing specific cases where the spectrum determines the graph and constructing examples of cospectral nonisomorphic graphs.

Contribution

It characterizes spectral determination of almost complete graphs under various edge deletion patterns and constructs nonisomorphic cospectral graphs for larger deletions.

Findings

Graphs with edges deleted as a matching or complete bipartite are spectrally determined.

Deletion of a path's edges results in graphs determined by their generalized spectrum.

For up to five deleted edges, only one pair of cospectral nonisomorphic graphs exists.

Abstract

We investigate when a complete graph $K_n$ with some edges deleted is determined by its adjacency spectrum. It is shown to be the case if the deleted edges form a matching, a complete graph $K_m$ provided $m \leq n-2$, or a complete bipartite graph. If the edges of a path are deleted we prove that the graph is determined by its generalized spectrum (that is, the spectrum together with the spectrum of the complement). When at most five edges are deleted from $K_n$, there is just one pair of nonisomorphic cospectral graphs. We construct nonisomorphic cospectral graphs (with cospectral complements) for all $n$ if six or more edges are deleted from $K_n$, provided $n$ is big enough.