TITLE: Does there exists a growing sequence of simple connected regular graphs of girth $k$ ($k \geq 5$) with uniformly bounded diameter?
QUESTION [5 upvotes]: Fix any $k \geq 5$.
Does there exist a growing sequence of simple connected REGULAR graphs, each of girth $k$, where the number of vertices $n$ goes to infinity and where the diameter of each graph is uniformly bounded in $n$?

REPLY [2 votes]: $\newcommand{\diam}{\operatorname{diam}}$
The answer is positive for girth $g\in \{5,6,7,8,12\}$.
To show that the graphs from the references are examples, we use the following arguments.
Below $d\ge 2$, $g\ge 3$ are natural numbers, and $G$ is a $d$-regular graph $G$ of girth $g$, order $n$, and diameter $\diam G$. Put $g’=\lfloor (g-1)/2\rfloor$.
Let $v$ be any vertex $v$ of $G$ let $B(u)$ be the set of vertices $u$ of $G$ such that the distance from $v$ to $u$ is at most $g’$.
$$|B(u)|=1+d+d(d-1)+\dots +d(d-1)^{g’-1}=1+\frac {d(d^{g’}-1)}{d-1}.$$
Denote the latter value by $b(d,g)$.
Lemma. $\left\lfloor 1+\frac{\diam G}{2g’+1}\right\rfloor b(d,g)\le n$.
Proof. Let $m=\diam G$. Pick vertices $v_0$ and $v_m$ of the graph such that the distance between them equals $m$ and let $v_0-v_1-\dots-v_m$ be a shortest path from $v_0$ to $v_m$. Then sets $B(v_0), B(v_{2g’+1}), B(v_{4g’+2}), \dots$ are mutually disjoint, otherwise we can make a shortcut in the path. It follows the lemma’s claim. $\square$
A $(d, g)$-graph is a $d$-regular graph of girth $g$. Erdős and Sachs [ES] proved the existence of $(d,g)$-graphs for all $d\ge 2$ and $g\ge 3$. Let $n(d,g)$ be the minimum order of $(d, g)$-graph. Lemma follows if $n(d,g)\le 2(d,g)-1$ then there exists a $(d,g)$-graph of diameter at most $2g’\le g$.
By Theorem 16 from [EJ], when $5\le g\le 8$ and $d$ is an odd prime power then $n(d,g)\le m(d,g)$. Also I expect a lot of constructions of $d$-regular graphs of girth $g$ and order at most $2b(d,g)-1$ is listed in [EJ], Section 4.1.2 (for girth $5$) and Section 4.1.3 (for girths $6$, $8$, and $12$). For instance, Construction XX by Gács and Héger shows that $n(d,12)\le m(d,12)$, when $d$ is a prime power. See also Section 4.1.4.
In [AFLN] and [AA-PBL2] are listed results on $n(d,5)\le 2b(d,5)-1$, see also [AA-PBL2, Proposition 2]. In [AA-PBL] are presented similar results for $g=7$ (Theorems 3.1 and 3.2) and $g=8$ (Theorems 3.3, 3.4, possibly, 3.5, and the references after it). Maybe some references from Introduction of [AA-PBL2] and Section 3.5.2 of [MS] can be helpful.
References
[AFLN] M. Abreu, M. Funk M., D. Labbate, V. Napolitano,A family of regular graphs of girth 5, Discrete Mathematics 308 (2008) 1810–1815.
[AA-PBL] M. Abreu, G. Araujo-Pardo, C. Balbuena, D. Labbate, A formulation of a $(q+1,8)$-cage.
[AA-PBL2] M. Abreu, G. Araujo-Pardo, C. Balbuena, D. Labbate, Families of small regular graphs of girth 5, Discrete Mathematics 312 (2012) 2832–2842.
[EJ] Geoffrey Exoo, Robert Jajcay, Dynamic Cage Survey, The electronic jornal of combinatorics, dynamic survey 16.
[ES] P. Erdős, H. Sachs, Reguläre Graphen gegebener Taillenweite mit minimaler Knotenzahl, Wiss. Z. Uni. Halle (Math. Nat.), 12 (1963) 251–257.
[MS] Mirka Miller, Jozef Širáň, Moore graphs and beyond: A survey of the degree/diameter  problem, The electronic jornal of combinatorics 20:2, dynamic survey 14v2.