32 problems

Let $G$ be a simple graph, let $n_G$ be its number of vertices, and let $S^V(G,\beta)$ denote its walk entropy at temperature $\beta\geq0$. The graph $G$ is walk-regular when every…

Let $G$ and $G_0$ satisfy the assumptions of the theorem preceding this statement, and let $h_G(p)$ and $h_{G_0}(p)$ denote their monomer–dimer entropies at density $p$. Entropy ma…

For each $r\geq 3$, let $B_r$ denote the lower-bound constant for rank-$r$ roses, and let $B_3$ be the rank-three constant. Monotonicity conjecture for $B_r$. … The claim would mak…

Let $\mathcal{R}_r$ denote the rank-$r$ rose, let $\mathcal{M}^1(\mathcal{R}_3)$ be the relevant space of normalized length functions, and let $B_3$ be the rank-three lower-bound c…

Let $G$ be a rank-$3$ graph with two vertices and four edges, and let $\ell=(\log(3),\log(3),\log(3),\log(3))$. Extremal graph conjecture. The most extreme case occurs for this gra…

For each rank $r\geq 3$, let $C_r$ be the constant from the theorem asserting that every rank-$r$ metric graph contains a proper subgraph whose entropy is at least $C_r$. Conjectur…

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 $n$ and diameter $d\ll n$. Let $I_{f_s}(T)$ and $I_{f_e}(T)$ be the distance-based entropies associated with the functionals $f_s$ and $f_e$. The fixed-d…

Let $T$ be a tree of order $n$, and let $I_{ecc}(T)$ denote its eccentricity-entropy. The diameter-three tree conjecture. Among trees of order $n$, the maximum value of $I_{ecc}$ i…

Let $G$ be a graph of order $n$, and let $I_{ecc}(G)$ denote its eccentricity-entropy. An extremal graph is one attaining the minimum value of this entropy. The minimum eccentricit…

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…

Strong subadditivity conjecture. The functional $F$ should also satisfy the strong subadditivity, equivalently submodularity, property for dependent random variables over ambiguous…

Let $G$ be a connected graph with order $n$ and size $m$, and let its degree sequence be written in nonincreasing order. The first-degree based graph entropy is the Shannon entropy…

Das–Dehmer conjecture. The path graph maximizes the first degree-based entropy among trees. This conjecture was eventually proved, so the extremal characterization is solved.

Let $G$ be a graph with degree sequence $(d_1,\ldots,d_n)$, and let $p_i=d_i/\sum_{j=1}^n d_j$ be its normalized degree sequence. The structural information of $G$ is the Shannon e…

Entropy conjecture. For $\alpha$ sufficiently large,

Let $G$ be a tree on $n$ vertices, and let $I_d^t(G)$ denote its $t$-th graph entropy based on vertex degrees, where $t$ is a real number: … with $d_i$ the degree of the $i$-th ver…

Let $G$ be a simple graph, let $n_G$ be its number of vertices, and let $S^V(G,\beta)$ denote its walk entropy at temperature $\beta>0$. Temperature-count conjecture. If $G$ is not…

Let $G$ be a simple graph, let $n_G$ be its number of vertices, and let $S^V(G,\beta)$ denote its walk entropy at temperature $\beta\geq0$. The graph $G$ is walk-regular when every…

Let $G$ be a simple graph, let $n_G$ be its number of vertices, and let $S^V(G,\beta)$ denote its walk entropy at temperature $\beta\geq0$. The graph $G$ is walk-regular when every…

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…

Kraus–Dehmer–Schaumann's conjecture. A minimal graph for $I_{f^\sigma}(G)$ is highly connected. In particular, a minimal graph on $n$ vertices has at least $m\geq n/2$ vertices of…

Kraus–Dehmer–Schaumann's conjecture. A graph minimizing $I_{f^2}(G)$ for the exponential sequence is a tree. Moreover, it is a generalized star of diameter approximately…

Dehmer–Kraus's conjecture. Among all trees $T_{n,d}$ with $d\ll n$, the two-tailed comet