25 problems

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

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…

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 $G=P_n$.

Let $C_{11}$ denote the cycle on eleven vertices. A Šoltés graph is a graph for which deleting any vertex leaves the Wiener index unchanged. Šoltés graph uniqueness conjecture. The…

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…

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…

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 $G_{m,n}=P_m\Box P_n$ 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 $(T,\sigma)$ 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 $e$,

Let $G$ be a maximal $5$-connected planar graph with $n\geq 12$ vertices. The 5-connected Wiener index conjecture. … This is presented as a conjectured sharp bound for $5$-connecte…

Wiener complexity and diameter conjecture. The Wiener complexity and the diameter of fullerene graphs of an arbitrary order having the maximal Wiener index are given in Proposition…

Maximal Wiener index conjecture. If a fullerene graph of an arbitrary order has the maximal Wiener index, then it is a nanotubical fullerene graph with caps of types $a$--$d$, and…

Let $G$ be a connected graph of order $n\geq k$, and let $\mu(G)$ denote its average distance and $\mu_k(G)$ its average Steiner $k$-distance. Dankelmann–Oellermann–Swart conjectur…

Let $\mathcal{T}_{n}$ denote the class of trees on $n$ vertices, and let $W[\mathcal{T}_{n}]$ be the set of their Wiener indices. Define $W^{\int}[\mathcal{T}_{n}]$ to be the larges…

Let $\mathcal{T}_{n}$ denote the class of trees on $n$ vertices, and let $W[\mathcal{T}_{n}]$ be the set of their Wiener indices. Cardinality conjecture. The cardinality of…

Let $G$ be a $2$-connected graph of order $n\ge 10$. Let $K_n$ denote the complete graph, and let $K_n^2$ and $K_n^{n-2}$ be the graphs obtained from $K_{n-1}$ by adding one vertex…

Let $G$ be a connected graph and let $B_1,\ldots,B_k$ be all its non-complete blocks, of respective orders $n_1,\ldots,n_k$. Define the Szeged–Wiener difference by…

Sills–Wang's balanced-ordering conjecture. The quadratic form $q(x)$ is maximized by this arrangement, where $|l_i-r_i|\le 1$ and $l_i+r_i=m_i$ for $i=2,\dots,s$.

Let $A_k=(a_{ij})$ be the $k\times k$ matrix with $a_{ij}=\frac{1}{2}|i-j|$, and let $I_k$ be the $k\times k$ identity matrix. Define $C_k(\lambda)=\det(A_k-\lambda I_k)$ to be the…

Let $G$ be a connected graph with $n \geq 4$ vertices, $m \geq n$ edges, and an odd cycle. The Hansen–Li–Liu conjecture. asserts that … Moreover, the bound should be best possible,…