Decomposition of Smith graphs in maximal reflexive cacti

The spectrum of a graph is the family of eigenvalues of its (0,1)(0,1) adjacency matrix. A simple graph is reflexive if its second largest eigenvalue λ2λ2 does not exceed 2. The graphic property λ2⩽2λ2⩽2 is a hereditary one, i.e. every induced subgraph of a reflexive graph preserves this property and that is why reflexive graphs are usually represented through maximal graphs. Cacti, or treelike graphs, are graphs whose all cycles are mutually edge-disjoint.The set of simple connected graphs characterized by the property λ1=2λ1=2, where λ1λ1 is the largest eigenvalue, is known as the set of Smith graphs. It consists of cycles of all possible lengths and some trees. If two trees T1T1 and T2T2 have such vertices u1∈T1u1∈T1 and u2∈T2u2∈T2 which, after their identification u1=u2=uu1=u2=u give a Smith tree, we say that that Smith tree can be split at its vertex uu into T1T1 and T2T2.It has turned out that several classes of maximal reflexive cacti can be described in the following way: we start from certain essential cyclic structure with two characteristic vertices c1c1 and c2c2, and then form a family of maximal connected reflexive cacti by splitting Smith trees, and by attaching their parts to c1c1 and c2c2. This way of decomposition of Smith trees leads to an interesting phenomenon of so-called pouring of Smith trees between two vertices.