674 problems

Let $G$ be a decorated unicyclic graph with cycle length parameter $g$, let $R_G$ be its curvature operator, and let $w^\ast$ denote the Perron metric. Write … and let $\varphi$ be…

Central-quotient eigengap conjecture. Either $k_{>1}=|G|-1$, or

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 connected $d$-regular graph with adjacency matrix $A$ and $n$ vertices. Let $\xi_1,\xi_2,\ldots,\xi_n$ be orthogonal eigenvectors of $A$, with the conditions stated b…

Let $G_{n,\ell}:=K_{\underbrace{n,n,\ldots,n}_{\ell}}$ be the complete multipartite graph with $\ell$ parts, each of size $n$, and let $Z_q(G)$ denote the $q$-analogue of the zero…

Let $G$ be a connected simple graph of order $n$, with adjacency eigenvalues $\lambda_1,\ldots,\lambda_n$. Define $s^{+}(G)=\sum_{\lambda_i>0}\lambda_i^2$ and…

For every connected graph $G$ on $n$ vertices, let $\lambda_1,\ldots,\lambda_n$ be the eigenvalues of its adjacency matrix, and define $s^+(G)=\sum_{\lambda_i>0}\lambda_i^2$ and…

Let $G$ be a graph with $m=|E(G)|$ edges, let $\rho(G)$ denote its adjacency spectral radius, and let $W_{2k+1}=K_1\vee C_{2k}$. Determine the sharp upper bound for $\rho(G)$, incl…

Let $G$ be a graph of order $n$, and let $EP(G)=(p_{ij})$ be its edge-path matrix, where $p_{ij}$ is the maximum number of pairwise edge-disjoint paths between distinct vertices…

For positive integers $\ell_1\leq \ell_2\leq \ell_3$, let $T(\ell_1,\ell_2,\ell_3)$ be the tree with a unique degree-three vertex $v$ such that…

For all connected graphs $G$ and $H$, if their distance matrices $D(G)$ and $D(H)$ are cospectral, meaning that $\operatorname{Spec}(D(G))=\operatorname{Spec}(D(H))$, then $G$ is b…

For every fixed integer $n\ge 2$, the family of undirected generalized pancake graphs $\{P(m,n)\}_{m\ge 2}$ is an expander family; equivalently, there exists a constant $c_n>0$ suc…

For integers $n>2$ and $k>1$, let $F_{n,k}(x)=\Phi_n(1+x+\cdots+x^k)\in\mathbb{Q}[x]$, where $\Phi_n$ is the $n$-th cyclotomic polynomial. Determine, exactly in terms of $n$ and…

Let $G$ be an $R$-disjoint graph. Write $B(G)$ for its bipartite part, and for each odd cycle $C$ of $G$, let $R(C)$ denote the subgraph associated with $C$. Determinantal factoriz…

Let $\mathsf{T}$ be the tree in Fig., and let $T$ be any nontrivial tree with root $r$. Define $G_1=T(r) \odot \mathsf{T}(4)$ and $G_2=T(r) \odot \mathsf{T}(7)$. Smith normal form…

Let $n\in\mathbb{N}$, and let $S,T\subseteq\mathbb{Z}_n$ be integral symbols, so that $\Gamma_S=Cay(\mathbb{Z}_n,S)$ and $\Gamma_T=Cay(\mathbb{Z}_n,T)$ are integral circulant graph…

Distinct-root conjecture. One has $\rho_1\neq\rho_2$. This asserts that distinct admissible parameter triples determine distinct spectral radii for the corresponding Type $5$ digra…

Let $K_{n,m}$ be the complete bipartite graph with parts of sizes $n$ and $m$, and let $K_2$ be the complete graph on two vertices. Write $Z_q(G)$ for the $q$-analogue of the zero…

Let $n\geq 7$ be odd and let $I$ satisfy … For a partition $\lambda$ of $n$, write $\lambda_\lambda^I$ for the eigenvalue associated with the corresponding irreducible representati…

Let $A(n,q)$ be the algebraically defined graph introduced in the paper, and let its second largest eigenvalue mean the second largest eigenvalue of its adjacency matrix. Nearly Ra…

Hypercube spectral conjecture. The simple random walk $P^s$ is unstable, all eigenvalues of the corresponding linearized curvature-flow matrix are real, and

Complete-graph spectral conjecture. The simple random walk $P^s$ is asymptotically stable, and the eigenvalues of $DF(P^s)$ are

Let $T$ be a tree with diameter $d$, and let $m_T[0,1)$ denote the number of Laplacian eigenvalues of $T$ less than $1$. Almost-all-trees lower-bound conjecture. Almost all trees h…

Let $T$ be a tree of order $n$, let $d$ denote its diameter, and let $m_T[0,1)$ be the number of Laplacian eigenvalues of $T$ in the interval $[0,1)$. Define … where…

Minimum-connectivity–maximum-diameter conjecture. If $\Gamma_n$ has asymptotically minimum algebraic connectivity, then its diameter is asymptotically maximum, namely