  For any generic immersion of a Petersen graph into a plane, the number of
crossing points between two edges of distance one is odd. The sum of the
crossing numbers of all $5$-cycles is odd. The sum of the rotation numbers of
all $5$-cycles is even. We show analogous results for $6$-cycles, $8$-cycles
and $9$-cycles. For any Legendrian spatial embedding of a Petersen graph, there
exists a $5$-cycle that is not an unknot with maximal Thurston-Bennequin
number, and the sum of all Thurston-Bennequin numbers of the cycles is $7$
times the sum of all Thurston-Bennequin numbers of the $5$-cycles. We show
analogous results for a Heawood graph. We also show some other results for some
graphs. We characterize abstract graphs that has a generic immersion into a
plane whose all cycles have rotation number $0$.
