I have copied a section from Bondy's Graph theory book:
The idea is to count the mappings between two simple graphs $G$ and
$H$ on the same vertex set $V$ according to the intersection of the
image of $G$ with $H$. Each such mapping is determined by a
permutation $\sigma$ of $V$, which one extends to $G=(V,E)$ by setting
$\sigma(G):=(V,\sigma(E))$, where
$\sigma(E):=\{\sigma(u)\sigma(v):uv\in E\}$. For each spanning
subgraph $F$ of $G$, we consider the permutations of $G$ which map the
edges of $F$ onto edges of $H$ and the remaining edges of $G$ onto
edges of $\bar H$. We denote their number by $|G\to H|_F$, that
is: $$|G\to H|_F:=|\{\sigma\in S_n:\sigma(G)\cap H=\sigma(F)\}|$$
I personally deciphered the notation |G->H|F as no. of distinct mapping of edges from a spanning subgraph F of G to H and the remaining edges (edges of G – edges of F) being mapped to H' taken together.
In short it would be most helpful if someone could explain what |G → H|F notation means, would be most helpful with a simple example graph.
And sorry if it is a trivial question because I am new to graph theory and the biggest problem that I mostly face is understanding what a theorem means because of the extensive use of mathematical symbols and notations and I mostly interpret wrong or miss out something.
Best Answer
Subscript $F$.
$G$ and $H$ are two graphs on the same vertex set.
$F$ s a subgraph of $G$.
This notation is defined in the link.
$$ |\;G\rightarrow H\;|_F $$ is the number of permutaitons of $G$ that map the edges of $F$ onto edges of $H$, and the rest of the edges of $G$ onto $\overline{H}$. I guess $\overline{H}$ is the complement of $H$ (on the same vertex set).