699 problems

Cheeger-constant Nordhaus-Gaddum conjecture. If $G\neq S_n$, then

Let $G$ be a bipartite graph with $n$ vertices. The minimum distance signless Laplacian spread conjecture. … Equality holds if and only if … This proposes that the balanced complet…

Conjecture on ternary strong nodal domains. For any eigenfunction $f$ of $H(n,3)$, $n\geq 1$, with eigenvalue $3n$ we have $\mathrm{SND}(f)\geq n+1$.

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 $G$ be a tree on $n\geq 2$ vertices. Write $E_{\mathrm{ISI}}(G)$ for its inverse sum indegree energy, and let $S_n$ and $P_n$ denote the star and path on $n$ vertices, respecti…

Let $n\ge 5$ and $m\ge n$. Write ${\mathcal T}(n,m)$ for the class of weighted trees on $n$ vertices with total edge weight $m$, and let the energy of a weighted tree be the sum of…

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 $G$ be a connected graph, and let $\rho(G)$ denote its spectral radius, $d$ its degree vector, and $\operatorname{var}(G)$ its degree variance. Define the degree deviation by…

The path-matrix conjecture. If $M$ is 2-connected, then its path matrix has exactly one positive eigenvalue.

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

For every integer $r\ge 3$ and every simple $r$-regular graph $G$ on $n=|V(G)|$ vertices, if $F(G)$ denotes the number of spanning forests of $G$, then…

For every $\varepsilon>0$, there exist constants $c_{\varepsilon}>0$ and $n_{0}(\varepsilon)$ such that, for every $n\geq n_{0}(\varepsilon)$ and every $n$-vertex graph $G$ that is…

For every regular triangle-free graph $G$ of order $n$, if $\lambda_1(G)\geq\cdots\geq\lambda_n(G)$ are the adjacency eigenvalues of $G$, then…

For every edge-color-critical graph $F$ with $\chi(F)=r+1$, and every positive integer $n$, let $\mathcal{G}_{n,r}(F)$ be the family of all $n$-vertex, $F$-free graphs that are not…

For any two graphs $G$ and $H$ of order $n$, with adjacency matrices $A_G$ and $A_H$, let $\mathbf e$ be the all-one vector and define…

For every $\varepsilon>0$, there exists a constant $c_{\varepsilon}>0$ such that, for every sufficiently large $n$, every integer $d\leq (1-\varepsilon)n/2$, and every $d$-regular…

Let $G$ be an $(n,d,\lambda)$-graph, meaning a $d$-regular graph on $n$ vertices whose nontrivial adjacency eigenvalues have absolute value at most $\lambda$. For every $\delta>0$,…

Let $\tau(G)$ denote the relaxation time of the simple random walk on a connected regular graph $G$ with $n$ vertices. Aldous and Fill conjectured that, as $n\to\infty$, … and that…

For integers $n\ge 3$ and $2\le \Delta\le n-1$, let $\lambda_1(n,\Delta)$ denote the maximum adjacency spectral radius among all connected nonregular graphs $G$ of order $n$ with m…

Determine the sharp quantitative local-to-global spectral bound for down-up walks on the faces of a pure simplicial complex, assuming sufficiently strong spectral expansion of the…

Let $k\ge 2$, $k\le \delta\le 2k-1$, and set $h=\delta-k$. For all sufficiently large $n$, among the graphs $G\in\mathcal{L}_{\mathcal{H}}^{2}(n,k,\delta)$, the graph maximizing th…

Let $\mathcal{C}_n^k$ denote the $k$-uniform hypercycle on $n$ vertices. A hypergraph is called integral if all of its adjacency eigenvalues are integers. The conjecture states tha…

Fix $\alpha>1$. Let $G_n\sim G(n,\alpha/n)$, let $B_n$ be its non-backtracking matrix, and order the eigenvalues by nonincreasing modulus, so that…