Limit for the Euler-genus distributions of ladder-like sequences of graphs

In this paper, we demonstrate a method for calculating the production matrices for the Euler-genus polynomials of H-linear families of graphs. Particularly, for any ladder-like sequence of graphs, we give its explicit production matrix that leads to a recurrence relation. Based on this recurrence relation, we show that the Euler-genus distributions of any ladder-like sequence of graphs are asymptotic to a normal distribution. Since the genus polynomials of the ladder-like sequence of graphs have already been calculated by Chen et al.(J Algebr Combin 52:137–155, 2020), their crosscap-number polynomials are also known.