44 problems

Elphick–Farber–Goldberg–Wocjan conjecture. Every connected graph $G$ on $n$ vertices satisfies

For a graph $G$, define its positive adjacency $p$-energy by … Let $p\ge2$ be a real number, and let $G$ be a connected graph with $n$ vertices. Tang–Liu–Wang's positive $p$-energy…

Let $R$ be a finite CC-ring, meaning that the centralizer of every non-central element of $R$ is commutative, and let $\Gamma_R$ be its commuting graph. A graph is CN-hyperenergeti…

Let $K_n$ be the complete graph, let $e$ be an edge of $K_n$, and let $\boldsymbol{\dot{A}}(K_n)$ and $\boldsymbol{\dot{A}}(K_n-e)$ denote the degree-based weighted adjacency matri…

Akbari–Alazemi–Anđelić's conjecture. For any connected graph $G \not\cong C_3, C_5, C_7$ with $\Delta(G) \in \{ 2, 3, 4, 5 \}$, we have

Let $G$ be an $n$-vertex graph, and let $\xi_f(G)$ denote the projective rank of $G$. Projective-rank energy conjecture. … This is proposed as a weakening of Fajtlowicz's graph ene…

Let $G$ be a graph, and let its diminished Sombor energy be the sum of the absolute values of the eigenvalues of its diminished Sombor matrix. Nonintegrality conjecture. There does…

A graph is maximal planar if it is planar and no edge can be added while preserving planarity, and maximal outerplanar if it is outerplanar and no edge can be added while preservin…

Let $G$ be a connected graph of order $n$, and let $\omega(G)$ denote its clique number. Clique-number refinement. If … then … This conjecture is motivated by the expectation that…

Let $G$ be a unicyclic graph of order $n$, and let $k$ be the odd length of its unique cycle. Unicyclic graph square-energy conjecture. … The conjecture is motivated by an exact ca…

Let $G$ be a connected graph of order $n$ and size $m$. Strong positive square-energy conjecture. If $m\ge n+1$, then … This conjecture strengthens the positive-energy side of the…

Let $G$ be a graph, let $p\geq 2$, and let $\chi_v(G)$ denote its vector chromatic number. Let $\mathcal{E}_p^+(G)$ and $\mathcal{E}_p^-(G)$ denote its positive and negative $p$-en…

Let $G$ be a connected graph with $n$ vertices, and let $\mathcal{E}_p^+(G)$ and $\mathcal{E}_p^-(G)$ denote its positive and negative $p$-energies. Elphick–Wocjan conjecture. For…

Positive $p$-energy path-minimization conjecture.

Nikiforov's extremal $p$-energy conjecture. For $1\leq p\leq2$,

Guo's monotonicity conjecture. Adding an edge should not decrease positive square energy:

For a graph $G$, let $\operatorname{surp}(G)$ denote its surplus, and let $s^-(G)$ denote its negative square energy. Negative surplus exponent conjecture. … The paper constructs e…

Let $G$ be a graph with $m$ edges, and let $s^+(G)$ denote its positive square energy. Positive square-energy exponent conjecture. … The paper proves the same lower bound with a do…

Akbari et al.'s conjecture. There exists a subset $S\subseteq V(G)$ such that

Let $\Gamma$ be a finite graph, and let $K_{|v(\mathcal{G})|}$ denote the complete graph on $|v(\mathcal{G})|$ vertices. A graph is hyperenergetic if $E(\Gamma)>E(K_{|V(\Gamma)|})$…

Let $G$ be a simple graph, let $V(G)$ be its vertex set, and for $S\subseteq V(G)$ let $G_S$ denote the graph obtained by adding loops at the vertices in $S$. Write…

Average-degree energy conjecture. Then

Akbari–Alazemi–Andjelić conjecture. The inequality

Let $p$ be a prime and let $\alpha$ be a positive integer. Consider the group-annihilator graph realised by the group $\mathbb{Z}/p^{\alpha}\mathbb{Z}$. Hypoenergeticity conjecture…

Let $G=(V,E)$ be a graph, with the energy game assigning each coalition its graph energy and with $\boldsymbol{\varphi}$ denoting the Shapley value. Energy-game core conjecture. Fo…