25 problems

Let $G$ be a finite simple connected graph of order $n=|V(G)|\ge2$. For each vertex $v$, let $d(v)$ be its degree and $N(v)$ its open neighbourhood, and define … Let $D(G)$ be the…

Let $T$ be a tree with $n$ vertices. For $0\leq k\leq n-2$, let $d_k(T)$ denote the normalized coefficient of degree $k$ in the distance characteristic polynomial of $T$. A sequenc…

Steiner distance hyperdeterminant value conjecture. The value of $\det(M)$ depends only on $n$ and $k$, not on the choice of the tree $T$.

Triangle inequality conjecture. For any pure states $\textup{\textbf{x}},\textup{\textbf{y}},\textup{\textbf{z}}\in\mathbb{P}(\mathbb{C}^n)$,

Let $G$ be an interval graph. A graph is called BDM-constructible if it is uniquely determined by its boundary distance matrix (with the relevant order and boundary fixed). Interva…

Let $G$ and $H$ be graphs, and suppose that a single similarity matrix establishes cospectrality of their generalized distance matrices for two distinct values of $q$, neither of w…

Coefficient formula conjecture. The coefficient of $x^{2n-5}$ is

Let $G$ be a connected graph with order $n$ and boundary $\partial(G)$. Its boundary distance matrix $\hat{D}_G$ records the distances between vertices of $\partial(G)$. Both $n$ a…

Let $G_1$ and $G_2$ be graphs with $B_1\subseteq V(G_1)$ and $B_2\subseteq V(G_2)$. Coalescing conjecture. If coalescing the same connected rooted graph onto every vertex of $B_1$…

Let $D_k(T)$ denote the Steiner distance hypermatrix of order $k$ associated with a tree $T$ on $n$ vertices. Steiner distance Graham–Pollak determinant conjecture. The quantity…

Let $G$ be a connected graph of order $n\geq 4$, with remoteness $\rho$, and suppose that $G\ncong K_n$ and $G\ncong K_n-e$. Here $K_n-e$ is the complete graph with one edge remove…

Let $G$ be a unicyclic graph, let $\Delta$ be its distance squared matrix, let $\ell$ be the number of leaves of $G$, and let $q$ be the number of negative eigenvalues contributed…

Let $H_1$ and $H_2$ be graphs, and let $B_1\subseteq V(H_1)$ and $B_2\subseteq V(H_2)$ be vertex sets. Let $\mathcal{D}$ denote the distance matrix, whose $(u,v)$-entry records the…

Let $X$ be an affinely independent subset of the Hamming cube $H_n=\{0,1\}^n$ with $n+1$ points, and let $D$ be its distance matrix for the Hamming metric. Inverse-distance-sum con…

Let $G$ be a connected graph on $n\geq 4$ vertices with diameter $D$, proximity $\pi$ and distance spectrum $\partial_1\geq \ldots \geq \partial_n$. Auchiche–Hansen's conjecture. ……

Let $T$ be a tree with $n\geq 3$ vertices. Let $d_k(T)$ denote the normalized coefficients of the distance characteristic polynomial, defined from the coefficients $\delta_k(T)$ by…

Let a level-$2$ network be a network whose blobs have level at most $2$, let a macaron be a level-$2$ blob with two cut-edges, and let an alt-path structure be the structure descri…

Let $G$ be a connected non-transmission-regular graph with $n$ vertices. Its distance spectral radius $\lambda_1(G)$ is the largest eigenvalue of its distance matrix, and…

Let $H_n$ be the Hamming cube, and let $D$ be the distance matrix of an affinely independent subset $\{x_0,x_1,\ldots,x_m\}\subseteq H_n$, where $m\geq 1$. Let $\mathbf{1}$ denote…

Let $n,k$ be positive integers, let $0<\alpha<1$, and let $G\in\mathcal{G}_{n,k}$. Here, $\mu_\alpha(G)$ denotes the generalized distance spectral radius of $G$, and $K(n,k,1)$ is…

Collins–Shor conjecture. The sequence $d_0(T),\ldots,d_{n-2}(T)$ is unimodal, with its peak between $\big\lfloor \frac n2\big\rfloor$ and $\big\lceil n-\frac n{\sqrt{5}}\big\rceil$…

For a tree degree sequence $\mathbf{d}$, let $Err(\mathbf{d})$ be the relative error between the upper and lower bounds for the minimum distance spectral radius. The six-percent er…

Let $\mathbf{d}$ be a tree degree sequence, and let $BFS(\mathbf{d})$ denote its breadth-first-search (greedy) tree. For a graph $G$, let $RD(G)$ be its terminal distance matrix, a…

Let a tree degree sequence be fixed, and let $BFS(\mathbf{d})$ denote the breadth-first-search (greedy) tree generated by it. The BFS-tree conjecture for distance spectral radius.…

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…