Assigning random orientation to an edge in a regular graph

Question

Given a simple regular graph of degree $d$ on $n$ vertices.

Assume an ordering of vertices and assume all orientations of edges is from $i$ to $j$ if edges $ij$ exists and $i<j$. Pick $m$ random edges and flip their orientation. For every simple closed path $p$ denote $\sigma(p)$ by difference in number of edges with orientation $i$ to $j$ with $i<j$ and number of edges with orientation $i$ to $j$ with $i>j$.

What is the probability that if we pick a random simple closed path $p$ then $\sigma(p)\bmod 2\equiv0$? How many such simple closed path can we expect?

At least can we tell anything about this when graph is complete?

Answer 1

If the length of simple closed path $P$ is equal to $l$ , I think $\sigma (p) \equiv 0 (mod2)$ is equivalent with $l \equiv 0 (mod2)$ . If the length of simple closed path $P$ is even , then $\sigma (p)$ is even , and conversely if the length of $P$ is odd , then $\sigma(p)$ is odd. Therefore it's enough to counting all cycles in the graph and all even cycles in the graph. The ratio of two number is wanted probability.