39 problems

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…

For $n\geq 5$, let $C^n_{1,2}$ be the circulant graph obtained from the cycle on $n$ vertices by adding the edges joining each vertex to the vertex two steps away, with indices tak…

For integers $n\geq 1$ and $k$, let $C^n_{1,k}$ be the graph on vertices identified with $\{1,\ldots,n\}$, obtained from the cycle by adding the edges $\{i,i+k\}$, with vertex labe…

Let $C_n(1,2)$ be the circulant graph on even $n$ vertices, let $A_\sigma$ range over its signed adjacency matrices, and define … The switching classes are coordinatized by triangl…

Lifting conjecture. Then, for some $S$,

Let $C_{n_1}(R_1)$ and $C_{n_2}(R_2)$ be circulant graphs such that … for some $R$. Product criterion conjecture. The graph $C_{n_1n_2}(R)$ has Type-2 isomorphic circulant graphs i…

Exact labeling-number conjecture.

Exact labeling-number conjecture.

Word-representability conjecture. Every 5-regular circulant graph is word-representable.

Let $\Gamma(n,S)$ be a circulant graph on $n$ vertices with connection set $S$. Here, “small” means that $S$ is a connection set whose size is small relative to $n$, although no pr…

Prism and Möbius ladder conjecture. Their Speyer polynomials are

Buratti-Horak-Rosa conjecture. If

Type I conjecture. Every such graph is of Type I.

Type II classification conjecture. If $\Gamma$ is nontrivially unstable and of Type II, then

Let $n$ be the order of a graph, and let a circulant graph be a graph whose vertices and adjacency relation are invariant under cyclic translation. Let $\operatorname{mp}(G)$ denot…

A cubic $8$-circulant nut graph is a cubic nut graph admitting an $8$-circulant structure. Nonexistence conjecture for cubic $8$-circulant nut graphs. There exists no cubic $8$-cir…

The conjecture. The graph $\Gamma$ is unstable if and only if either there exists a nonzero $h\in 2\mathbb{Z}_{2m}$ such that

A circulant nut graph is a circulant graph whose adjacency matrix has a one-dimensional null space spanned by a full vector. Let $d$ be a degree satisfying $d \equiv_4 0$, and let…

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…

Let $n$ be an integer, and let a circulant graph mean a graph whose vertices are arranged cyclically with adjacency determined by a fixed set of cyclic differences. The monophonic…

For integers $n$ and $k$ with $k\leq n/2$, let $\operatorname{Circ}(n,k)$ denote the circulant graph with these parameters. A graph is a prime distance graph if its vertices can be…

Campos–de Mello's conjecture. If $G=C_n^k$, then

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 $S=\{a_1,a_2,\dots,a_6\}$…