7 problems

Let $G$ be a simple graph, and let $\delta(G)$ denote its minimum degree. The maximum nullity $\mathrm{M}(G)$ is the greatest nullity among matrices associated with $G$ in the maxi…

Greaves–Yatsyna's sharpness conjecture. For every fixed $e\geqslant 3$ and all sufficiently large $n$, this bound is sharp.

Let $v$ be an $n$-dimensional integral vector whose entries are all nonzero modulo $p$. Suppose that … Here $e$ denotes the all-one vector, and a vector generates a primitive matri…

Let $S(n,k)$ denote the family of $n\times n$ zero-one matrices with every row and column sum equal to $k$, and let $M(n,k)$ be the maximum absolute determinant of a matrix in…

Let $R_t(s)$ denote the maximum proportion of invertible $2$-by-$2$ submatrices among the relevant submatrices of an invertible binary matrix of size $s$ for parameter $t$, and sup…

Rokicki et al.'s conjecture. Based on computational results for $n\leq 120$, one has

Let $A$ be a maxdet $\\{\pm 1\\}$ matrix of order $n$, and let $M$ range over all $m\times m$ submatrices of $A$, with $n\geq m$. Write $E(\det(M)^2)$ for the mean value of…