64 problems

Gutman et al.'s conjecture. For any graph $G$ of order $n$,

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…

Nikiforov's conjecture. If $G$ is a connected graph of order $n$ and $1<p<2$, then

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…

For a graph $G$, let $A(G)$ be its adjacency matrix and define its energy by … Here $G=G(n)$ has order $n$, and $\sigma_i(A(G))$ are the singular values of its adjacency matrix. Ko…

For a graph $G$, define its negative adjacency $p$-energy by … Let $G$ be a connected graph on $n$ vertices and let $p\ge2$. Akbari–Kumar–Mohar–Pragada's negative $p$-energy conjec…

Let $G$ be a graph with no isolated vertex, let $A_G$ be its adjacency matrix, and let $(n^+,n^0,n^-)$ be the inertia of $A_G$, consisting of the multiplicities of its positive, ze…

Let $P_n^{6,6}$ denote the bicyclic molecular graph of order $n$ corresponding to the alpha,beta diphenyl-polyene $C_6H_5(CH)_{n-12}C_6H_5$. Gutman–Vidović's conjecture. For $n=14$…

Caporossi–Cvetković–Gutman–Hansen conjecture. If $G$ has minimum energy among all connected graphs with $n\ge 6$ vertices and $m$ edges, then $G$ is $O_{n,m}$ when

Akbari–Kumar–Mohar–Pragada–Zhang conjecture. If $k\equiv 3\pmod 4$, then $s^+(G)>n>s^-(G)$; if $k\equiv 1\pmod 4$, then $s^+(G)<n<s^-(G)$. This predicts that the residue of the odd…

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 $r$ and $s$ be odd, and let $\ell$ be odd. For a complex unit gain dumbbell graph with gain parameters $(\alpha,\beta)$ and energy $E(\Phi)$, the conclusion that the minimum of…

Let $G$ and $H$ be simple, undirected, and connected graphs, and let $e(G)$ denote the graph energy of $G$. Two graphs are locally equienergetic when they have equal graph energy.…

Universality conjecture. For every pair of distinct odd primes $(p,q)$, the set $\mathcal{D}^*$ is the unique maximiser of $E(p^2q^3,\mathcal{D})$ over all admissible $\mathcal{D}$…

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…

Let $G$ be a complete $k$-partite graph, and let $e$ be an edge of $G$ such that $G\backslash\{e\}$ remains connected. The distance energy of a graph is the sum of the absolute val…

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 $n$. Positive square-energy equality conjecture. If … then $G$ is a bipartite unicyclic graph. The conjecture is proposed as a possible equali…

Let $G$ be a connected graph of order $n$. Square-energy equality conjecture. … This conjecture concerns the equality cases in the main positive square-energy conjecture. The sourc…

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…