23 problems

The Asymptotic Upper Matching Conjecture. Under these hypotheses,

The Asymptotic Lower Matching Conjecture. Under these hypotheses,

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 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…

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 $G$ 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…

Entropy conjecture. For $\alpha$ sufficiently large,

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 $G$ 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. Randić–degree-power entropy distance conjecture. It holds … Here $R$ is the Randić index and $If_1$ is the degree-power entropy…

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

Dehmer–Kraus–Schaumann's conjecture. The star graph has the minimal value of $I_{f^3}(D)$.

Dehmer–Kraus's conjecture. For every sequence

Chen–Dehmer–Shi's conjecture. The balanced double star and the comet $CS(n,t_0)$ attain the maximum and minimum values of $I_{f^n_2}(T)$, respectively.

Cao and Dehmer's conjecture. For $l>0$, $I_{f^l_d}(G)$ is a monotonically increasing function of $l$ for connected graphs.

Cao–Dehmer–Schaumann's conjecture. We have