70 problems

Let $C_n(A)$ and $C_n(B)$ be circulant graphs, where $A$ and $B$ are inverse-closed subsets of $\mathbb Z_n\setminus\{0\}$. For a unit $t\in U(n)$, multiplication by $t$ gives an i…

Wilson's conjecture. These four sufficient conditions should also be necessary for a circulant graph to be nontrivially unstable.

Buratti-Horak-Rosa conjecture. If

Let $K_n$ be the complete graph on $n$ vertices, and let edge lengths be the integers from $1$ through $\lfloor n/2\rfloor$, with the length of an edge defined by the cyclic distan…

Let $G$ and $H$ be circulant graphs whose eigenvalues are all integers. So's conjecture. The graphs $G$ and $H$ are isomorphic if and only if they are cospectral. This conjecture c…

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…

Consider the constructed families of undirected circulant graphs of dimension $5$, with degrees $10$ and $11$, whose orders are largest known and are given by quintic polynomials d…

Let a saturated graph be a regular Eulerian graph whose avoidance index equals its degree, and let a circulant graph be a graph defined by a generating set on a cyclic vertex group…

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…

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

Ghebleh–Niepel conjecture. In the cases not covered by their exact-value results, the lower bound in each inequality should be increased by one, so that the corresponding upper-bou…

McKay's conjecture. The identity holds for all even orders, for every $n$ and $r$. It is known for square-free $n$ and has been verified for all orders less than $54$; the conjectu…

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…

Let $n$ and $d$ be positive integers, and let $\operatorname{Circ}(n,\{1,2,\ldots,d\})$ be the directed circulant graph whose arcs correspond to the generators $1,2,\ldots,d$. Let…

Exact labeling-number conjecture.

Exact labeling-number conjecture.

Khennoufa–Togni conjecture. Only finitely many graphs $C_n(1,3)$ are Type II; all remaining graphs $C_n(1,3)$ are Type I.

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

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…