The equator of a graph is the length of a longest isometric cycle. We bound the order $n$ of a graph from below by its equator $q$, girth $g$ and minimum degree $\delta$ - and show that this bound is sharp when there exists a Moore graph with girth $g$ and minimum degree $\delta$. The extremal graphs that attain our bound give an analogue of Moore graphs. We prove that these extremal `Moore-like' graphs are regular, and that every one of their vertices is contained in some maximum length isometric cycle. We show that these extremal graphs have a highly structured partition that is unique, and easily derived from any of its maximum length isometric cycles. We characterize the extremal graphs with girth 3 and 4, and those with girth 5 and minimum degree 3. We also bound the order of $C_4$-free graphs with given equator and minimum degree, and show that this bound is nearly sharp. We conclude with some questions and conjectures further relating our extremal graphs to cages and Moore graphs.

翻译：图的赤道定义为最长等距环的长度。我们通过图的赤道$q$、围长$g$和最小度$\delta$来下界估计其阶数$n$，并证明当存在围长为$g$、最小度为$\delta$的Moore图时，该下界是紧的。达到我们给出的界的极值图可视为Moore图的一种推广。我们证明这些极值“类Moore”图是正则的，且其每个顶点都包含在某个最大长度的等距环中。我们进一步证明这些极值图具有高度结构化的划分，该划分是唯一的，并且可以从任意一个最大长度等距环轻松导出。我们刻画了围长为3和4的极值图，以及围长为5且最小度为3的极值图。我们还估计了给定赤道和最小度的$C_4$-free图的阶数，并证明该界几乎是紧的。最后，我们提出了一些问题和猜想，以进一步探讨这些极值图与笼图及Moore图之间的关系。