Explain a graph homomorphism in terms of group theory

Question

I am taking a course in group theory right now, and a homomorphism from $(G_1, *)$ to $(G_2, \circ)$ has been defined as a function $h: G_1 \to G_2$ such that $h(a * b) = h(a)\circ h(b)$. I have also seen the definition of graph homomorphisms, and I am wondering how they translate to the language of group theory? That is, what are $G_1$ and $G_2$ in this case, and what are $*$ and $\circ$?

Answer 1

The general idea of a homomorphism is just that it preserves the "algebraic structure" of its domain in its codomain: it preserves the structure that the a binary operation provides for a group, and it preserves the structure on a graph's set of vertices, namely their adjacency. I suppose symbolically we could define a relation $\mathrm{R}$ on a graph's set of vertices by saying $x\mathrm{R}y$ if if there is an edge from $x$ to $y$. Then a graph homomorphism $f\colon G \to G'$ would be a map such that for all vertices $a,b$ of $G$, we have $$ a\mathrm{R}b \implies f(a)\mathrm{R}f(b)\;. $$