The star characterization conjecture for graphs with one large Laplacian eigenvalue

Let $G$ be a finite simple graph with Laplacian eigenvalues $\mu_1\geq\cdots\geq\mu_n$ and average degree $\overline{d}(G)=2m/n$, where $m$ is the number of edges and $n$ the number of vertices. Define $\sigma(G)$ to be the number of Laplacian eigenvalues at least $\overline{d}(G)$. For integers $r\geq 1$ and $s\geq 0$, let $K_{1,r}+sK_1$ denote the disjoint union of the star $K_{1,r}$ and $s$ isolated vertices. Star characterization conjecture. $\sigma(G)=1$ if and only if $G$ is isomorphic to $K_1$, $K_2+sK_1$ for some $s\geq 0$, or $K_{1,r}+sK_1$ for some $r\geq 2$ and $0\leq s<r-1$. The conjecture seeks a complete structural characterization of graphs having exactly one Laplacian eigenvalue at least their average degree; the paper proves it for several classes, including graphs whose complements are disconnected, but the general case remains open.