How to construct a 5-regular graph with diameter 2 on 22 vertices?

Here's one:

$$(1, 2), (1, 4), (1, 6), (1, 8), (1, 11), (2, 10), (2, 12), (2, 17), (2, 21), (3, 4), (3, 8), (3, 17), (3, 18), (3, 20), (4, 9), (4, 10), (4, 22), (5, 6), (5, 9), (5, 13), (5, 18), (5, 21), (6, 14), (6, 16), (6, 20), (7, 8), (7, 9), (7, 11), (7, 12), (7, 16), (8, 19), (8, 21), (9, 12), (9, 15), (10, 16), (10, 18), (10, 19), (11, 13), (11, 15), (11, 18), (12, 14), (12, 17), (13, 17), (13, 19), (13, 22), (14, 18), (14, 19), (14, 22), (15, 19), (15, 20), (15, 21), (16, 20), (16, 22), (17, 20), (21, 22)$$

I obtained this via integer linear programming as follows. Let $N=\{1,\dots,22\}$ be the nodes, and let $P=\{i\in N, j\in N: i<j\}$ be the set of node pairs. For $(i,j)\in P$, let binary decision variable $x_{i,j}$ indicate whether $(i,j)$ is an edge. For $(i,j)\in P$ and $k \in N \setminus \{i,j\}$, let binary decision variable $y_{i,j,k}$ indicate whether $k$ is a common neighbor of $i$ and $j$. The constraints are: \begin{align} \sum_{(i,j)\in P:\ k \in \{i,j\}} x_{i,j} &= 5 &&\text{for $k\in N$} \tag1\\ x_{i,j} + \sum_{k \in N \setminus \{i,j\}} y_{i,j,k} &\ge 1 &&\text{for $(i,j)\in P$} \tag2\\ y_{i,j,k} &\le [i<k]x_{i,k} + [k<i]x_{k,i} &&\text{for $(i,j)\in P$ and $k \in N \setminus \{i,j\}$} \tag3\\ y_{i,j,k} &\le [j<k]x_{j,k} + [k<j]x_{k,j} &&\text{for $(i,j)\in P$ and $k \in N \setminus \{i,j\}$} \tag4 \end{align} Constraint $(1)$ enforces $5$-regularity. Constraint $(2)$ enforces diameter $2$. Constraints $(3)$ and $(4)$ enforce that $y_{i,j,k}=1$ implies $k$ is a neighbor of $i$ and $j$, respectively.