11 problems

The paper considers simple triangulations and quadrangulations of a given order and connectivity, where proximity is a distance measure on a connected graph and the stated construc…

Let $S$ be a simple random walk on $\mathbb{Z}^{4}$ started at the origin. For $0\leq m\leq n$, let ${\cal G}_{m,n}$ be the random graph formed by the sites visited by $S$ from tim…

Let $S$ be a simple random walk on $\mathbb{Z}^{d}$ started at the origin. For $0\tsymm m\leq n$, let ${\cal G}_{m,n}$ be the random graph whose vertices are the sites visited by…

Let $a\from (2,5/2)$, let $\bigl(f_i\bigr)_{i\geq 1}$ be the sampled marked vertices or faces, let $d^\dagger_\mathrm{gr}$ denote the graph distance, and let…

For a positive integer $d$, a distance-$d$ packing of cycles in a graph $G$ is a set of cycles such that no path of length at most $d$ joins two distinct cycles. For a vertex set…

Let $P_n$ be the path with $n$ vertices, and let $b[?25l\pi(P_n)$ denote the smallest positive value of the total relative displacement among permutations of $V(P_n)$. Chartrand et…

Typical-distance universality conjecture. The random variable $f(Z)$ concentrates at some value $\mathcal{D}$.

Let $T$ and $T'$ be any two trees with $n$ vertices. Energy–graph-entropy distance conjecture. It holds … Here $E$ is the graph energy and $Ig$ is the entropy based on the absolute…

Let $T$ and $T'$ be any two trees with $n$ vertices. Wiener–Randić distance conjecture. It holds … The conjecture compares graph distance measures induced by the Wiener and Randić…

Let $G$ be a connected graph. For a vertex $v$ in $G$, let $ecc_G(v)=\max\{d_G(v,x)\mid x\in V(G)\}$, let $diam(G)$ denote the diameter of $G$, and define the boundary of $G$ by ……

Let $d(x,y)$ denote the graph distance between vertices $x,y\in\mathbb{Z}^d$, with $d(x,y)=\infty$ when $y\notin\mathcal{C}(x)$. Let $\tau=\beta\alpha/d$, and suppose that $\tau>2$…