If $G$ be any graph , then what is the upper bound of the number of cycles in $G$ if I add one edge to $G$?
it is easy to visualize that adding one edge to $G$ it may not form any cycle (for this if we take one isolated vertex then adding an edge to it does not form any cycle). But is there be any upper bound of it after adding exactly one edge?
I think after adding one edge the no of cycles in $G$ is increased by atmost 1. So, is there be anything?
There is no upper bound on the number of cycles created by the new edge. If for example a graph G has two cycles $xe_1e_2...y...e_k$ and $xe'_1e'_2...y...e'_l$, and $xy \notin E(G)$, then adding the edge $xy$ creates four new cycles $$xe_1e_2...y$$ $$xe_ke_{k-1}...y$$ $$xe'_1e'_2...y$$ $$xe'_le'_{l-1}...y$$ With this example it is easy to see that if we considered 3 cycles, adding the edge would add 6 new cycles, etc. If the graph had $n$ of these cycles and we added the edge we would create $2n$ new cycles.