Is it purely coincidental that the same number $2\sqrt{d-1}$ appears in these two following apparently disparate concepts?

A $d-$regular graph is said to be called Ramanujan if its adjacency eigenvalues except the highest and the lowest are inside the interval, $[-2\sqrt{d-1}, 2\sqrt{d-1}]$. (one considers only one copy of the highest and the lowest if either has multiplicities)

The largest root of the matching polynomial of a graph with largest degree is $d$ over its vertices, is $2\sqrt{d-1}$

A side question : anyone knows of a pedagogic rewriting/exposition of the second result apart from its original proof in this paper, http://projecteuclid.org/euclid.cmp/1103857921 ?

A related fact that has been shown recently is that if one assigns elements of $\mathbb{Z}_k$ to the set of oriented edges of a graph ( such that the group element assigned to he edge $(u,v)$ is inverse of the group element assigned to $(v,u)$ ) then over all such signings $s$, one has, $\mathbb{E}_s [ det ( xI - A_{s,i} ) ] = \mu (x)$ where $A_{s,i}$ is the $i-$fold Hadamard product of the signed adjacency matrix $A_s$ and $\mu$ is the matching polynomial of the graph.