17 problems

Let $\mathcal P_m(n)$ be the generalised pancake graph and let $Q(\mathfrak P_{m,n})$ be the associated matrix used in the paper. For a symmetric matrix or graph, write $\lambda_1$…

For a graph $G$, let $\tau_k(n)$ denote the maximum, over graphs on $n$ vertices, of the sum of the $k$ largest eigenvalues of $G$, and let … for fixed $k\geq 1$. The asymptotic Ky…

Let $G$ be any graph of order $n$, and let $\lambda_1(G)$ and $\lambda_2(G)$ denote its two largest adjacency eigenvalues. Ebrahimi–Mohar–Nikiforov–Ahmady conjecture. The spectral…

Let $\mathcal{T}(n)$ be the family of trees on $n$ vertices. For a tree, write $\lambda_1$ and $\lambda_2$ for its two largest adjacency eigenvalues, and let $DC(k,k,\ell)$ denote…

Let $\mathcal P_m(n)$ denote the generalised pancake graph, and let $\lambda_1(\mathcal P_m(n))$ and $\lambda_2(\mathcal P_m(n))$ be its largest and second-largest signless Laplaci…

Let $G$ be a graph of order $n$, and let $\lambda_{n-1}$ and $\lambda_n$ denote its two smallest eigenvalues. Strengthened eigenvalue conjecture. One has … This strengthens the ori…

Lin–Miao–Guo conjecture. If $1/2\leq\alpha<1$, then

Structural conjecture. If $d\ge 5$ is odd, then the following hold for $T^\#$:

Elphick–Liu–Ning conjecture. For any connected graph,

Let $G$ be a graph with maximum degree $\Delta(G)>1$. An edge-signing of $G$ is a map from $E(G)$ to $\{-1,1\}$; write $G^\sigma$ for the resulting edge-signed graph and let…

Abdi–Ghorbani's uniqueness conjecture. For every $n\geq11$, the $n$-vertex graph of ${\cal F}_{\rm qua}$ is the unique graph with minimum spectral gap among connected quartic graph…

Let $G$ be a connected graph, and let $u$ and $v$ be adjacent vertices of $G$ with degree at least $2$. Let $G_{p,q}(u,v)$ denote the graph obtained by attaching pendent paths of l…

Let $G$ be a connected graph, let $u$ be a vertex of $G$, and let $G_{p,q}(u)$ denote the graph obtained by attaching pendent paths of lengths $p$ and $q$ at $u$. For…

Let $G$ be a connected graph on $n$ vertices with $\frac{1}{2}<\alpha<1$. Star-minimization conjecture. … with equality if and only if $G\cong K_{1,n-1}$. This conjecture asks whet…

Alazemi–Andelić–Simić conjecture. In any chain graph, every vertex is downer with respect to every non-zero eigenvalue. Equivalently, for every chain graph $G$, every $v\in V(G)$,…

Strict asymptotic bound conjecture. There exist infinitely many integers $k$ such that

Componentwise inertia sum-of-squares conjecture. One has