9 problems

Let $k\ge 2$ be an integer, and define … For a graph $G$, write $D$ for its diameter. Miklavič–Šparl conjecture. For every $n>n_k$, the generalized Petersen graph $GP(n,k)$ is not…

Let $n\ge 3$ and $1\le k<n/2$. The generalized Petersen graph $GP(n,k)$ is defined by … … For a connected graph $G$, vertices $x$ and $y$ are balanced when $|W_{xy}|=|W_{yx}|$, whe…

Jerebic's conjecture. For any $k\ge 2$, there exists a positive integer $n_0$ such that $GP(n,k)$ is not distance-balanced for every $n\ge n_0$. The conjecture was positively resol…

Let $k\geq 2$, and let $GP(n,k)$ denote the generalized Petersen graph with parameters $n$ and $k$. A graph is distance-balanced if, for every pair of vertices $u$ and $v$, the num…

A distance-balanced graph is a graph in which, for every pair of vertices $u$ and $v$, the numbers of vertices closer to $u$ than to $v$ and closer to $v$ than to $u$ are equal. A…

Let $\Gamma$ be a graph. For vertices $u,v\in V(\Gamma)$, let $W_{u,v}$ denote the set of vertices closer to $u$ than to $v$, and define $d(u,S)=\sum_{x\in S}d(u,x)$ for a vertex…

Let $G$ be a quasi-$\lambda$-distance-balanced graph. For positive integers $k_{1}$ and $k_{2}$, call $G$ $(k_{1},k_{2})$-regular when … and no adjacent vertices have the same degr…

Let $G$ be a quasi-$\lambda$-distance-balanced graph. For a vertex $u$ of $G$, write $\deg(u)$ for its degree and $D(u)=\sum_{v\in V(G)}d(u,v)$ for its total distance (transmission…

The eventual non-$\ell$-distance-balance conjecture. For any $n>n_k$, the graph $GP(n,k)$ is not $\ell$-distance-balanced for any integer $\ell$ with $1\leq\ell<D$. Moreover, $n_k$…