Non-convexity of the eigenvalue region of standardized Laplacians and the case n = 4 of the Agaev–Chebotarev conjecture

Let $\Lambda_n$ be the set of all eigenvalues of $n\times n$ standardized Laplacian matrices, the real matrices with zero row sums and off-diagonal entries in $[-1/n,0]$. In 2005 Agaev and Chebotarev constructed a polygon $S_n\subseteq\Lambda_n$ and conjectured that $\Lambda_n=S_n$. Chebotarev recently showed that the conjecture fails for every $n\geq5$, by a second-order perturbation of circulant matrices at the vertices of $S_n$. We prove that $\Lambda_4=S_4$, so that the conjecture holds exactly for $n\leq4$, and that $\Lambda_n$ is not convex for $n\geq5$. For $n=4$ a linear programming argument reduces the problem to digraphs with at most six fractional weights, and the resulting polynomial inequality of degree six, which is attained with equality on a three-parameter family, is verified on each of $9583$ classes by exact Bernstein expansions. For $n\geq5$ every point of $\Lambda_n$ near the vertex $\lambda_1$ lies within a quadratic distance of the line through the edge $[\lambda_1,\lambda_2]$, which is incompatible with the second-order displacement; hence $\Lambda_n$ is not even star-shaped with respect to $\lambda_1$. We also show that at the horizontal edges of $S_n$, $n$ even, no $C^2$ curve of standardized Laplacians through the corresponding circulant moves the eigenvalue outward at second order, and we give explicit digraphs with integer weights for $n=5$ and $n=6$ whose eigenvalues lie at distances $2.47\cdot10^{-3}$ and $1.79\cdot10^{-3}$ beyond $S_n$.