32 problems

Let $A$ and $B$ be non-negative $N\times N$ matrices. Write $A\circ B$ for their Hadamard product, $AB$ for their usual matrix product, and let $\rho$ denote the spectral radius. Z…

Let $G=(V(G),E(G))$ be a simple graph, let $v$ be a non-isolated vertex of $G$, and let $G-v$ be the graph obtained by deleting $v$ and all edges incident to it. Write $\rho(G)$ fo…

Daubechies–Lagarias conjecture. The generalized spectral radius and the joint spectral radius are equal for any finite set of matrices.

Let $X$ be a Fano manifold, let $QH^*(X)$ be its quantum cohomology ring, and let $c_1$ denote the first Chern class of $X$. The quantum product by $c_1$ defines a linear operator…

Let $G$ be a connected graph and let $u,v\tnobreak\text{ be vertices with }d(u),d(v)geq 2$. Suppose that $u$ and $v$ are joined by a path … where $d(w_i)=2$ for $1\leq i\leq s-1$.…

Bochi–Fayad conjecture. The set of all resistant pairs $(H,R)\in\mathcal{H}\times\mathcal{R}$ has full Lebesgue measure. Partial results are known, and explicit resistant pairs can…

Let ${\boldsymbol{M}}_{l}$ be the matrices associated with the user-placement scenario in which ${\boldsymbol{M}}_{l}$ is not an inverse Z-matrix for every $l\in\{1,\ldots,N\}$. Al…

Symmetrized-matrix convexity conjecture. Convexity of $l_{\boldsymbol{M}}$ is linked to the positive semidefiniteness of ${\boldsymbol{M}}+{\boldsymbol{M}}^t$.

For a $g$-tuple of matrices $\mathcal X=(X_1,\dots,X_g)$, define its outer spectral radius by … where $\operatorname{rad}(X)$ is the ordinary spectral radius. Let…

Let the random self-similar set be generated on the line by a homogeneous iterated function system whose common contraction ratio is the reciprocal of a non-negative integer and wh…

Let $A=A_s+A_d\varepsilon$ be an $n\times n$ dual number matrix and let $k\in\mathbb{N}$. Here $O$ denotes the zero matrix, and $\rho(A_s)$ denotes the spectral radius of the stand…

Let $A=A_s+A_d\epsilon$ be an $n\times n$ dual number matrix, where $A_s$ is its standard part, $A_d$ is its dual part, and $\rho(A_s)$ denotes the spectral radius of $A_s$. The ze…

Let $I=[0,1]$ be the domain, let $S$ be the vertical scaling function, and let $M_k$ and $M'_k$ be the matrices associated with the $k$-th subdivision, with spectral radii denoted…

Let $\mathcal B$ be a nonnegative fourth-order PS-tensor, and let the upper bound in Theorem denote the bound established there for its M-spectral radius $\rho_M(\mathcal B)$. Tigh…

Let $p=p_n$ satisfy $p_n\to\infty$, and let $\mathbf{z}_1,\ldots,\mathbf{z}_{p_n}$ be the eigenvalues of the truncated circular unitary matrix. Assume that the truncation parameter…

Spectral-radius convergence conjecture. In the critical case of finiteness of second moments, one should have

Local limit asymptotic conjecture. There exists a constant $c(x,y,\lambda)>0$ such that

Finite-variance no-outlier conjecture. The spectral radius of $P_n$ is bounded above by $\sigma+o(1)$ in probability as $n\to\infty$, assuming only that the atom variables…

Finite-variance spectral-radius conjecture. The spectral radius of $\frac{1}{\sqrt{n}}X_n$ converges to $1$ in probability as $n\to\infty$, assuming only that $\xi$ has mean zero a…

Bordenave–Caputo–Chafaï–Tikhomirov conjecture. The convergence in probability

Asarin's conjecture. The equality above should be valid for certain classes of non-negative matrices.

Let $M_2(\mathbb{R})$ be the space of all $2\times2$ real matrices, and let … be the lower spectral radius as a function on pairs of matrices. Discontinuity-dimension conjecture. T…

Let $A_n$ be a square random matrix with independent and identically distributed centered entries of variance $1/n$ and finite fourth moment, and let its spectral radius be the max…

Strict exclusion conjecture. One has

For a finite set $A\subset\mathbb{R}^+$, define … For $k\geq 1$, define … A cubic minimum conjecture asserts … where $T(k)=k(k+1)/2$ is the $k$-th triangular number; moreover,…