# Binary operations on graphs

Are there "binary operations" on graphs like in (https://en.wikipedia.org/wiki/Graph_product), which make the set of all graphs ("under consideration")

• a (abelian) group or
• a (commutative) ring or
• a field or
• some other algebraic structure ? By a graph I mean everything which might be considered a graph, for example a directed graph, undirected graph, graphs with no multiple edges, weighted graphs etc. If so, is there a reference on how the binary operations are construced? I managed to find

http://link.springer.com/article/10.1007/s40590-015-0081-7

and

http://www.sciencedirect.com/science/article/pii/S1571065314000092

but don't have access to it.

• You should require the operations to be natural in some sense; better yet would be to specify an appropriate sense of "natural". Otherwise, the answer is that any nonempty set admits binary operations making it a commutative ring. – Andreas Blass May 21 '16 at 20:24
• All graphs do not form a set, they form just a class.But the groups etc. usually are defined to be sets with operations. – Ilya Bogdanov May 21 '16 at 20:33