For integers $k,g,d$, a $(k;g,d)$-cage (or simply girth-diameter cage) is a smallest $k$-regular graph of girth $g$ and diameter $d$ (if it exists). The order of a $(k;g,d)$-cage is denoted by $n(k;g,d)$. We determine asymptotic lower and upper bounds for the ratio between the order and the diameter of girth-diameter cages as the diameter goes to infinity. We also prove that this ratio can be computed in constant time for fixed $k$ and $g$. We theoretically determine the exact values $n(3;g,d)$, and count the number of corresponding girth-diameter cages, for $g \in \{4,5\}$. Moreover, we design and implement an exhaustive graph generation algorithm and use it to determine the exact order of several open cases and obtain -- often exhaustive -- sets of the corresponding girth-diameter cages. The largest case we generated and settled with our algorithm is a $(3;7,35)$-cage of order 136.

翻译：对于整数 $k,g,d$，一个 $(k;g,d)$-笼图（或简称为围长-直径笼图）是具有围长 $g$ 和直径 $d$ 的最小 $k$-正则图（若其存在）。$(k;g,d)$-笼图的阶记为 $n(k;g,d)$。我们确定了当直径趋于无穷大时，围长-直径笼图的阶与直径之比的渐近下界和上界。我们还证明了对于固定的 $k$ 和 $g$，该比值可以在常数时间内计算。我们从理论上确定了 $n(3;g,d)$ 的精确值，并统计了 $g \\in \\{4,5\\}$ 时对应的围长-直径笼图的数量。此外，我们设计并实现了一种穷举图生成算法，并用其确定了若干开放案例的精确阶数，并获得了——通常是穷举的——对应围长-直径笼图集合。我们通过算法生成并解决的最大案例是一个阶为 136 的 $(3;7,35)$-笼图。