46 problems

Let $G$ be a finite connected graph, let $d$ be its diameter, let $W(G)$ denote its Wiener index, and let $C_{2d+1}$ denote the cycle on $2d+1$ vertices. DeLaViña–Waller conjecture…

For a graph $G$, let $V(G)$ denote its vertex set, let $G\setminus v$ be the graph obtained by deleting $v\in V(G)$, and let $W(G)$ be the Wiener index of $G$. A Šoltés' graph is a…

Let $G_{m,n}=P_m\Box P_n$ be the Cartesian product of paths on $m,n\geq 2$ vertices. Let ${D}_{m,n}$ be the orientation of $G_{m,n}$ with all $P_m$-layers oriented up except the la…

Let $G$ be a connected graph, and let $W(G)$ denote its Wiener index and $\varepsilon(G)$ its total eccentricity. For a graph with a prescribed number of vertices, paths are the gr…

Center–median conjecture. The center and median of $T_0$ have distance zero:

Let $D=(d_1,d_2,\dots,d_{t-1},d_t,1,\dots,1)$ be a degree sequence with $d_t\geq 2$ and $d_2\neq 2$, and let $G$ be a unicyclic graph with degree sequence $D$. Define … and let…

Hriňáková–Knor–Škrekovski conjecture. For large $n$ and $k\geq 2$, among all graphs $G$ on $n$ vertices, $R_k(G)$ attains its maximum at $G=K_n$ and its minimum at $G=P_n$.

Hriňáková–Knor–Škrekovski conjecture. For sufficiently large order $n$, the path $P_n$ has the smallest value of $R_2$ among all trees on $n$ vertices.

Let $G$ be a graph, and let $W(G)$ denote its Wiener index. A Šoltés graph is a graph for which deleting any vertex leaves the Wiener index unchanged. A graph is vertex transitive…

A Šoltés graph is a graph $G$ for which deleting any vertex leaves the Wiener index unchanged. Regularity conjecture. If $G$ is a Šoltés graph, then $G$ is regular. This conjecture…

Knor–Škrekovski–Tepeh conjecture. If $D$ maximises $W(D)$ among all orientations of $T$, then there exists a vertex $v$ in $D$ such that for every vertex $u$ there exists either a…

For integers $n,k,j$, let $G_{n,k,j}$ be the graph formed from a path $P_k$ and a clique $K_{n-k}$ by joining one end vertex of the path to $j$ vertices of the clique. Let a broom…

For integers $n,k,j$, let $G_{n,k,j}$ be the graph formed from a path $P_k$ and a clique $K_{n-k}$ by joining one end vertex of the path to $j$ vertices of the clique. Let $I_w$ de…

Let $T$ be a tree of order $n\geq 5$, and let $S_n$ denote the star on $n$ vertices. The Wiener-entropy $I_w(T)$ is defined from the vertex transmissions of $T$. The star conjectur…

For a graph $G$, let $W_*(G)$ be its minimal signed Wiener index, namely the minimum of $W_\sigma(G)$ over all possible signings $\sigma$. Let $T$ be an $n$-vertex tree and let…

A signed graph is a graph whose edges are assigned positive or negative signs. For a signed graph $(G,\sigma)$, let $W_\sigma(G)$ denote the sum of the signed distances over all un…

Let $\mathcal{T}$ be a tree, let $\mathcal{O}$ be a primitive growth operation, and consider the formula for the Wiener index of the tree resulting from applying $\mathcal{O}$ to…

Let $T$ be an $n$-vertex tree. Define the minimum signed Wiener index by … where the minimum ranges over all signings $\sigma$ of $G$. Let $\mathcal{D}$ be the set of all $n$-verte…

Let $(T,\sigma)$ be a signed tree on $n$ vertices. For the path $P_n$, let $+$ denote the constant signing that assigns $+1$ to every edge, and let $\alpha$ denote the alternating…

Wiener–eccentricity monotonicity conjecture. For every such edge $e$,

Path extremal conjecture. The difference between the Wiener index and the eccentricity satisfies

Let $G$ be an Eulerian graph of order $n$, meaning that every vertex of $G$ has even degree. For $n\geq 5$, let $C_{n,3}$ be the graph obtained from the disjoint union of cycles on…

Let $G$ be a $delta$-regular graph of order $n$, meaning that every vertex of $G$ has degree $delta$. Let $\mu(G)$ denote the average distance between unordered pairs of vertices o…

Czabarka et al.'s conjecture. The value $W(Q_n)$ is an upper bound for the Wiener index of quadrangulation graphs of order $n$.

Czabarka et al.'s conjecture.