7 problems

Let $\mathbb{G}_n=G(\mathbf{Y}^{(n)},\mathbf{W}^{(n)},\kappa_n)$ satisfy the assumptions of the local weak convergence theorem in the source, except for the assumption imposing reg…

Let $G_n$ be the random graph and let $o_1,o_2$ be the two vertices whose distance is considered. Assume the hypotheses of Theorem. Then typical distance lower-bound conjecture. Th…

For positive integers $\Delta$ and $t$, let $h_t(\Delta)$ be the smallest integer such that every graph $G$ with at least $h_t(\Delta)$ edges and maximum degree…

For positive integers $\Delta$ and $t$, let $h_t(\Delta)$ be the smallest integer such that every graph $G$ with at least $h_t(\Delta)$ edges and maximum degree…

Let $G$ be a graph, let $\alpha(G)$ be its independence number, and let $r(G)$ be its graph radius. Graffiti's radius conjecture. … The source verifies the intended bound for the l…

Let $G$ be a graph, let $\alpha(G)$ be its independence number, and let $\bar d$ be the average distance between distinct vertices of $G$. Graffiti's average-distance conjecture. ……

Let $\mathbb{G}_n$ and the uniformly chosen vertices $U_{n,1},U_{n,2}$ be as in Theorem … , there is a constant $\varepsilon_{\ast}\in(0,1)$ depending on the model parameters such…