Cayley graphs are just group gossip
Handed a group $G$ and a symmetric generating set $S$, you get a graph for free.
For a group $G$ and symmetric $S \subseteq G$ (so $s \in S \Rightarrow s^{-1} \in S$), the Cayley graph $\mathrm{Cay}(G,S)$ has vertex set $G$ and an edge $\{g,h\}$ whenever $g^{-1}h \in S$.
The setup
Adjacency is a one-liner: $A_{g,h} = 1 \iff g^{-1}h \in S$. Symmetry of $S$ makes the graph undirected and $|S|$-regular.
Why every vertex looks the same
Left-multiplication by $a \in G$ sends $g \mapsto ag$ and preserves adjacency, since $(ag)^{-1}(ah) = g^{-1}h$. So $G$ acts on its own Cayley graph by automorphisms, transitively.
Stand on any vertex and the neighbourhood looks identical. That is the property every averaging argument quietly leans on.
A tiny example
Take $\mathbb{Z}_n$ with $S=\{\pm 1\}$ and you have drawn a cycle $C_n$:
def adjacent(g, h, S, n):
return (h - g) % n in S # additive group: g^{-1}h is h - g
Where the spectrum comes in
For abelian $G$ the eigenvalues are character sums, $\lambda_\chi = \sum_{s \in S}\chi(s)$, so you read the spectral gap straight off the generators — and the gap controls mixing and expansion.