22 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 $\lambda_1$…

Let $G$ be a graph of order $n$, and let $f701_1(G)$ and $f701_2(G)$ denote its largest and second-largest adjacency eigenvalues, respectively. The Ebrahimi–Mohar–Nikiforov–Ahmady…

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 $T^\#$:

Inertia lower-bound conjecture. For any graph $G$ with inertia $(n^+,n^0,n^-)$,

Question on negative quadratic spread. Is there a family of graphs on $n$ vertices, with $n\to\infty$, such that

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

van Dam–Kooij's conjecture. For $D=n-e$ and $n$ sufficiently large, the open quipu $OQ_{n,e}$ is one of the graphs minimizing the spectral radius among graphs with $n$ vertices and…

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 $T$ be a tree with $n$ vertices, and let $d=2-\frac{2}{n}$ be its average degree. Trevisan's conjecture. The number of eigenvalues of $T$ smaller than $d$ is at least … The con…

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 $v\in V(G)$,…

Let $G$ be a connected graph with $n$ vertices and $m$ edges. Let $mu_1\geq\cdots\geq\mu_n$ be the eigenvalues of its adjacency matrix, and let $s^+$ and $s^-$ be the sums of the s…

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

Componentwise inertia sum-of-squares conjecture. One has

Positive-eigenvalue sum-of-squares conjecture. For every connected graph,

Maximum-spread conjecture. The maximum spread $s(n)$ of the graphs of order $n$ is attained only by $G(n,\lfloor 2n/3\rfloor)$; equivalently,