What is the dimensionality of Euclidean space where $N$-star can be embedded? How many dimensions are required to embed a star graph with $N$ vertices such that the following holds:

*

*The embedding has unit radius i.e. $\| x_0 - x_i \| = 1$ where $x_0$ is the central vertex, $i \neq 0$.

*$\| x_i - x_j \| \geq 1, i \neq 0, j \neq 0, i \neq j$.

basically, its a question about the number of points one can fit on a hypersphere such that the distance between any two points is $\geq$ than the radius of the hypersphere.
Seems like it is related to the graph dimension?
To give some simple examples, a 2-star graph (2 leaves) can be embedded in 1 dimension. A 3-star graph requires 2 dimensions.
 A: It seems reasonable to first answer this question to solve this problem:

What is the maximum number of points that can be placed on a sphere of
$\mathbb{R}^n$ of radius $1$ such that the minimum distance between them
is at least $1$?

In turn, this problem is equivalent to the problem about the kissing number
In the general case this problem is unsolved. Here is what is known today (November 2022)
for the first $10$ dimensions:
$$
\begin{array}{|c|l|}
\hline
\rm dim  & \rm number\ kissing\\
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  1 & 2   \\
  2 & 6   \\
  3 & 12  \\
4  & 24 \\
5  & 40-44\\
6  & 72-78\\
7  & 126-134\\
8  & 240\\
9  & 306-363\\
10 & 500-553\\
  \hline
\end{array}
$$
We see that already for $\mathbb{R}^5$ only the lower $(40)$ and upper $(44)$ bounds  are known, but the number itself is unknown.
Thus, one can place with the specified property in $\mathbb{R}^2$ but not in $\mathbb{R}$ those $k$-stars for which  $3\leq k\leq6$.
And furthermore:
$\mathbb{R}^3$, $7\leq k\leq12$;
$\mathbb{R}^4$, $13\leq k\leq24$
and so on.
Here is a similar question.
