cycle

let $\text{[math]}$ be a permutation. let $\text{[math]}$ , and let $\text{[math]}$ be the smallest positive integer so that $\text{[math]}$ (which we know exists by permutation.html). then we say that the entries $\text{[math]}$ form an $\text{[math]}$ -cycle in $\text{[math]}$ .
[cite:;taken from @combinatorics_bona_2023 definition 6.5]

a negative cycle in a graph is a cycle whose edges are such that the sum of their weights is a negative value.

negative cycles prevent some challenges when dealing with weighted graphs, as they can form cycles with a weight of $\text{[math]}$ .

consider the following graph whose weights function is taken from the real number line
$\text{[math]}$ at first glance the distance from $\text{[math]}$ to $\text{[math]}$ seems to equal 10, but if we were to go back and forth from the node $\text{[math]}$ to $\text{[math]}$ we'd find that on each step the distance decreases by -2, which would mean that the distance equals $\text{[math]}$ , this is why when dealing with negative weights we only consider directed graphs

a cycle $\text{[math]}$ is clockwise if it corresponds to a dual cut $\text{[math]}$ such that $\text{[math]}$ where $\text{[math]}$ . a cycle $\text{[math]}$ is counterclockwise if it corresponds to a dual cut $\text{[math]}$ such that $\text{[math]}$ where $\text{[math]}$ . this is demonstrated in broken link: blk:fig-cycle-1.

broken link: xopp-figure:/home/mahmooz/brain/pen/1738425953.4321527.xopp