7 problems

Let $n$ be the order of a graph, and let a nut graph be a graph whose adjacency matrix is singular with one-dimensional null space spanned by an eigenvector having no zero entries.…

Let $\mathcal{D}(n,t)$ denote the graph used in the construction, let $F_3$ and $F_5$ be the graphs shown in the paper, and let $\operatorname{\Box}$ denote the Cartesian product o…

Let $n\geq 9$ be a composite integer. A nut graph is a graph whose adjacency matrix has nullity one and whose nullspace is spanned by a vector with no zero entries; write $o_v(G)$…

Let $t$ be an integer and let $n$ be the order of a circulant graph. A circulant nut graph is a circulant graph whose adjacency matrix has nullity one and whose non-zero null-space…

Let $n$ be the order of a circulant graph, and let $\mathrm{Circ}(n,S)$ denote the circulant graph with generator set $S$. A nut graph is a non-trivial graph whose adjacency matrix…

Universal generator-set conjecture. For each odd $t\geq 3$, there exists $p_t$ such that $\operatorname{Circ}(n,\{1,\dots,2t+1\}\setminus\{p_t\})$ is a nut graph for every even…

A circulant nut graph $C(n,S)$ is a circulant graph on $n$ vertices whose adjacency matrix has nullity one and whose nullvector has no zero entries; here $S=\{a_1,a_2,\dots,a_6\}$…