A class of reflexive cactuses with four cycles

A simple graph is reflexive if its second largest eigenvalue λ2 is less than or equal to 2. A graph is a cactus, or a treelike graph, if any pair of its cycles (circuits) has at most one common vertex. For a lot of cactuses the property λ2 ≤ 2 can be tested by identifying and deleting a single cut-vetex (Theorem 1). if this theorem cannot be applied to a connected reflexive cactus and if all its cycles do not form a bundle, such a graph has at most five cycles. On the same conditions, in this paper we find some classes of maximal reflexive cactuses with four cycles. The complete case of four cycles, together with that of five cycles, is being settled in [10].