On existence and regularity of graphs with certain properties

In the first part we consider such graphs of finite diameter d and finite girth g in which any two vertices of distances are connected by the same number of distinct paths of the length =< d. These graphs are shown to be regulated or many values of d.g and s. In the second part we study the properties of graphs with g=<2d+1 in which all pairs of distinct vertices are connected by exactly t=1 distinct paths (for t=1 we get the so-called Moore graphs) We are concerned mainly with the question of existence for various values of d and t.