I'd like to preface this by saying this is the tip of the iceberg for an optional question for a summer REU program application, so if you think asking this question is in bad taste, let me know and I will remove it. This is not the question itself, but a related question which may show me some insight into how to tackle the actual question.
Let $W_4$ be the wheel graph consisting of four vertices. That is, $W_4$ is a graph consisting of a triangle and a central hub. The question is, what is the minimal number $m$ such that I can embed $W_4$ in $\mathbb R^m$ with the property that every edge of the graph is of length 1. Mostly I don't know how to begin to approach this problem, so I'm seeking some strategy for the problem, some relevant theorems that might be helpful, or really just anything that might give insight into how to approach this particular embedding.
The actual question for the REU is to generalize this statement to arbitrary wheel graphs, as well as the cylinder formed by adding and connecting another vertex to each non-hub vertex of the graph so that the edges are still unit $1$.
$W_4$ is also $K_4$ of course. However the answer is straightforward; the tetrahedron provides an example that $m=3$ is feasible, and arranging the vertices in two back-to-back equilateral triangles demonstrates that $m=2$ is not feasible as the last edge cannot be added. $m=3$ is also feasible (and probably required) for $W_5$ and $W_6$, for example pyramids of progressively lower height, and of course $m=2$ is possible for $W_7$.
I believe that $m=3$ is the highest dimension requirement, although the shapes will become more complex. You could probably satisfy yourself of this by constructing a chain of top-vertex-linked equilateral triangles out of card. I can easily imagine configurations with nested and adjacent attached pyramids allowing up to $W_{25}$ and I don't immediately see any fundamental barriers to continuing to clutter up 3D space around the central vertex.
Without clutter, we can make ring of fan-folded triangles around the central vertex, each edge showing the opposite fold to the previous. For an odd number of triangles (even $n$ in $W_n$) we can leave one triangle "flat". This allows extension to any $W_n$ only using $\mathbb{R}^3$.