Reliability polynomials crossing more than twice

In this paper we study all-terminal reliability polynomials of networks having the same number of nodes and the same number of links. First we show that the smallest possible size for a pair of networks that allows for two crossings of their reliability polynomials have seven nodes and fifteen edges. Then we propose a construction of pairs of graphs whose reliability polynomials exhibit an arbitrary number of crossings. The construction does not depend on multigraphs. We also give concrete examples of pairs of graphs whose reliability polynomials have three, four and five crossings, respectively, and provide the first example of a graph with more than one point of inflection in (0,1).