We study the game of Cops and Robbers, where cops try to capture a robber on the vertices of a graph. Meyniel's conjecture states that for every connected graph $G$ on $n$ vertices, the cop number of $G$ is upper bounded by $O(\sqrt{n})$, i.e., that $O(\sqrt{n})$ suffice to catch the robber. We present several families of abelian Cayley graphs that are Meyniel extremal, i.e., graphs whose cop number is $O(\sqrt{n})$. This proves that the $O(\sqrt{n})$ upper bound for Cayley graphs proved by Bradshaw is tight up to a multiplicative constant. In particular, this shows that Meyniel's conjecture, if true, is tight to a multiplicative constant even for abelian Cayley graphs.
In order to prove the result, we construct Cayley graphs on $n$ vertices with $Ω(\sqrt{n})$ generators that are $K_{2,3}$-free. This shows that the Kövári, Sós, and Turán theorem, stating that any $K_{2,3}$-free graph of $n$ vertices has at most $O(n^{3/2})$ edges, is tight up to a multiplicative constant even for abelian Cayley graphs.
New results on the energy of integral circulant graphs
循环图是并行分布计算中一类重要的互连网络。积分循环图在量子自旋网络的建模中扮演着重要的角色，它支持完美的态转移。整数循环图$\ICG_n (D)$具有顶点集$Z_n = \{0, 1, 2, ..., n - 1\}$，并且如果$\gcd(a-b,n)\in D$，则顶点$a$和$b$相邻，其中$D \subseteq \{d : d \mid n,\ 1\leq d<n\}$。这些图是高度对称的，具有积分谱和一些连接化学图论和数论的显著性质。图的能量最早由Gutman定义为图的邻接矩阵的特征值的绝对值之和。近年来，对具有相等能量的非共谱图对和族的研究有了很大的发展。在[R。B. Bapat，S. Pati，\textit{Energy of a graph is never an odd integer}，Bull. Kerala Math.Assoc.1（2004）129- 132.]，我们刻画了模4整循环图的能量。此外，我们还建立了整循环图能量的一般封闭表达式，推广了[A. Ili ' c，\textit{The energy of unitary Cayley graphs}，Linear Algebra Appl.431（2009），1881- 1889.]。最后，我们提出了一些开放的问题，并在整循环图的$n$顶点，如果$n$是偶数的图中，具有最小能量的极图的特征。
