Disjoint unions of complete graphs characterized by their Laplacian spectrum

International audienceA disjoint union of complete graphs is in general not determined by its Laplacian spectrum. We show in this paper that if we only consider the family of graphs without isolated vertex then a disjoint union of complete graphs is determined by its Laplacian spectrum within this family. Moreover we show that the disjoint union of two complete graphs with $a$ and $b$ vertices, $\frac{a}{b}>\frac{5}{3}$ and $b>1$ is determined by its Laplacian spectrum. A counter-example is given when $\frac{a}{b}=\frac{5}{3}$