Lubotzky-Phillips-Sarnak(LPS88) constructed Ramanujan graphs of degree $d=p+1$ (for an odd prime $p$) as Cayley graphs of projective linear groups with respect to a carefully chosen set of generators obtained using integer quaternions of norm $p$.
At a high level, we start with a free group $Gp(S)$ where $|S|=p+1$ is a generating set (including inverses). Our aim is to define appropriate relations (group presentation) based on our requirement so that we get a finite group and its Cayley graph.
The way LPS88 does this is as follows: interpret each element of $S$ as an integer (Lipschitz) quaternion $a+bi+cj+dk$ such that $a>0$ is odd, and $$a^2+b^2+c^2+d^2=p$$ By Lagrange's theorem, there are exactly $p+1$ solutions, and $Gp(S)$ can then be recovered as the multiplicative set, denoted $\Lambda$, in $L$ (the ring of Lipschitz integer quaternions) of these quaternions, multiplicatively modulo $p$. So far so good.
The advantage here is that we have embedded the free group generators in a ring, allowing us more structure and tools. For instance, LPS88 now go additively modulo $q$ (for another prime $q$) to get a finite ring of quaternions over $\mathbb{F}_q$ such that the multiplicative set $\Lambda$ in $L$ is mapped to invertible elements in the multiplicative group of this quotient ring. At this stage, we simply mod out powers of $p$ again, to get a finite group and the Cayley graph. The group can be shown to be a $2 \times 2$ projective linear group over the finite field $\mathbb{F}_q$.
Apologies for such a hurried and imprecise summary, but this was only to highlight the larger picture approach used. The details regarding why this is Ramanujan, etc. are not the focus of my question. In essence, LPS88 constructs the relations by embedding the generators in the ring of quaternion integers and going to a quotient ring, whose multiplicative group is then recovered. Its a roundabout way of defining relations, but is invaluable in keeping track of path/cycle length information.
My question is whether the reverse direction also makes sense. That is, suppose we are given a Cayley graph of a group $G$ and and a symmetric generating set of size $p+1$. That is, suppose we are given a group presentation: a set of $p+1$ generators and a set of relations.
Can we construct this Cayley graph in the LPS88 way? More precisely, is there an ideal $N$ in $L$ with $N \cap \Lambda = \phi$ such that the group $G$ is the image of $\Lambda$ in the quotient $L/N$ modulo the multiplicative subset generated by the image of $p$?
So is it possible that any given Cayley graph can be constructed by this method? Leave aside Ramanujan-ness and other properties. I am just curious if this framework can be used to construct arbitrary $p+1$-regular Cayley graphs.
If you have any clarifications, I'll edit and update the question accordingly. I'll also try to add some commutative diagrams soon to make it clearer what is needed. Thanks.