I consider three islands represented by vertices V and the travel routes by ship are represented by the edges E. Here G=(V,E).
I consider the non-empty set E and define the binary operation $\oplus_t$ called Travel. If I do not move at all then it is given by $0$ which acts like the additive identity. Travel is defined by the following:
$$x \oplus_t y = z$$ $$x \oplus_t z = y$$ $$y \oplus_t z = x$$ $$y \oplus_t x = z$$ $$z \oplus_t y = x$$ $$z \oplus_t x = y$$
The operation travel makes E into a group (abelian group). This seems fine to me... but still I thought of verifying.
Besides, I would like to add more graph theory questions to this matter and maybe a couple of group theory questions. Any suggestions? (The questions could be based on matter found in Graph theory- Derek Allan Horton and John Clark).
More generally, for an arbitrary graph, you can consider all paths starting at a particular node; this "path space" has some nice properties. By restricting to LOOPS that start and end at some particular node, you get the loopspace of the graph; there's a natural kind of "multiplication" in which you traverse one loop and then the other; the composition starts and ends at the basepoint, and hence is also a loop. There's no "inverse" in this situation though: you can't add something to a loop to make it the zero-step loop (the identity element). So it's a semigroup, and in general is nonabelian.
If your graph has some additional structure, like a filled-in triangle that you can "slide across", then you get a sense of equivalent loops -- called "homotopic loops" -- and the equivalence classes of loops have the same "multiplication" operation, except now there are inverses as well...and you've invented "the fundamental group of a finite simplicial complex", which is the starting point of many combinatorial topology books.
All this is to say: you're doing something interesting here, and tinkering with ideas like these isn't a bad thing at all!