A graph with degree set {r,r + 1} is said to be semiregular. A semiregular cage is a semiregular graph with given girth g and the least possible order. First, an upper bound on the diameter of semiregular graphs with girth g and order close enough to the minimum possible value is given in this work. As a consequence, these graphs are proved to be maximally connected when the girth g ≥ 7 is odd. Moreover an upper bound for the order of semiregular cages is given and, as an application, every semiregular cage with degree set {r,r + 1} is proved to be maximally connected for g є {6,8}, and when g = 12 for r ≥ 7 and r ≠ 20. Finally it is also shown that every ({r,r + 1};g)-cage is 3-connected.