7 problems

Let $\Gamma$ be an abelian group, and let $\mathbb{B}(\Gamma)$ be the Boolean algebra generated by the subgroups of $\Gamma$. A Cayley graph $\operatorname{Cay}(\Gamma,S)$ is integ…

Let $n\geq 4$ be an integer and set … The star graph is $\operatorname{Cay}(S_n,S)$. Abdollahi–Vatandoost's conjecture. The star graph is integral, and its set of distinct eigenval…

Let $n\geq 3$ be an integer, let $D(n)$ denote the positive divisors of $n$, and let $D\subseteq D(n)\setminus\{n\}$. Write $\operatorname{ICG}(n,D)$ for the integral circulant gra…

Let $n=p_1^{\alpha_1}\cdots p_s^{\alpha_s}\geq 3$ be the canonical prime factorization, with $p_1<\cdots<p_s$, and let $\mathcal{E}_{\min}(n)$ denote the minimum energy among integ…

For integers $n\geq 2$ and $d\geq 1$, define the unitary finite Euclidean graph $T_n^{(d)}$ by … Here $\mathbb{Z}_n^{\times}$ is the set of units of $\mathbb{Z}_n$. Li–Vinh's conje…

Let $G$ be a finite abelian group, let $1\notin S\subseteq G$ satisfy $S=S^{-1}$, and let $X(G,S)$ be the Cayley graph with vertex set $G$ in which $a$ and $b$ are adjacent when…

Let $\Gamma$ be a finite abelian group and let $S\subseteq\Gamma$ satisfy $-S=S$. The undirected Cayley graph $\operatorname{Cay}(\Gamma,S)$ has vertex set $\Gamma$, with vertices…