The number of paths in uniform cactus chains

A graph G is called g-gonal cactus if G has two end blocks and every block of G is Cg, a cycle of g vertices. This paper aims to establish the generating function and the recurrence relation to count the total number
of paths of all g-gonal cacti. Surprisingly, all g-gonal cacti (even the random structure) have the same total number of paths. By analyzing the Laurent series of the generating function, we derived the asymptotic formula for the total number of such paths as well. Furthermore, we constructed the formulae to calculate the number of paths of a given length of regular g-gonal cacti. The formulae are implemented in Python and provided in this paper.