4 problems

Let $\mathbb G$ be a simple directed graph with $n$ vertices and $m$ arcs, and let $L$ be its Laplacian matrix. Write the eigenvalues of $L$ as $0,\lambda_2,\ldots,\lambda_n$, and…

Nishikawa–Motter conjecture. Among all simple directed graphs with $n$ vertices and $m$ arcs, the minimum possible value of $\sigma^2$ is achieved when the Laplacian spectrum follo…

Geodesic-distance conjecture. The difference between the $1{,}000$-node and $10{,}000$-node lattice networks is caused by the average geodesic distance: the average node distance f…

Let $n$ be the number of nodes, let $m$ be the number of links, and define $q_k=k(n-1)$. For a network with Laplacian eigenvalues counted with multiplicity, let $\sigma_{\min}(m)$…