We are looking for the graphs with minimal detour index in the class of connected bicyclic graphs. For the fixed number of vertices, we split the problem into two cases: bicyclic graphs without common edges between cycles and the complement of it. In both cases, we find graphs with minimal detour index.

We consider the group action of the automorphism group Un = Aut(Zn) on the set Zn, that is the set of residue classes modulo n. Clearly, this group action provides a representation of Un as a permutation group acting on n points. One problem to be solved regarding this group action is to find its cycle index. Once it is found, there appears a vast class of related enumerative and computational problems with interesting applications. We provided the cycle index of specified group action in two ways. One of them is more abstract and hence compact, while another one is basically procedure of composing the cycle index from some building blocks. However, those building blocks are also well explained and finally presented in very detailed fashion.

The number of Boolean threshold functions is investigated. A new lower bound on the number of n-dimensional threshold functions on a set {0, 1,. .. , K − 1} is given.