On a class of tricyclic reflexive cactuses

Abstract

A simple graph is reflexive if the second largest eigenvalue of its (0, 1) adjacency matrix does not exceed 2. A graph is a cactus, or a treelike graph, if any pair of its cycles (circuits) has at most one common vertex. The subject of this paper is the class of tricyclic cactuses in which the central cycle is a quadrangle touching the rest two cycles at its non-adjacent vertices. In this class we describe a set of maximal reflexive graphs. The so-called "pouring" of Smith trees plays the crucial role in characterizing the resulting set.

On unicyclic reflexive graphs

“…In the same way as it was done in [5], we can verify the fact that a graph obtained by removing c 4 from the case (b) also allows no extension at the vertices of S i and S i (i = 1, 2). As for vertices of S 3 and S 4 , no extension is possible for the same reason as at free cycles in the case (a).…”

“…Of course, this includes attachment of the complete tree T , rooted at any vertex v, to v 1 and v 2 . Pouring of Smith trees turns out to be a very important tool in describing some classes of maximal reflexive graphs ( [5], [9], [10], [12], [13]). The result of the next Lemma was already used in [10].…”

“…7(d), where two Smith trees, S 1 · S 1 and S 2 · S 2 , pour between the vertices c 2 and c 3 . It was established in [5] (Lemma 5) that such a graph has λ 2 = 2 and that any extension by a pendant edge at any vertex of S 1 , S 2 , S 1 or S 2 implies λ 2 > 2. In order to construct a class of maximal unicyclic reflexive graphs, we should only examine the possibilities of extension at the vertices of the free cycle attached to c 1 .…”

“…8(a)), such a graph still has λ 2 = 2 and is a maximal tricyclic reflexive graph. It cannot be extended at vertices of two pouring Smith trees because of (the already mentioned) Lemma 5 of [5], while any extension at vertices of the free cycles is impossible because the removal of c 2 and application of Theorem RS to c 3 would give λ 2 > 2. Can the two free cycles be replaced by two Smith trees S 3 and S 4 ?…”

“…They correspond to some sets of vectors in the Lorentz space R p,1 and have some applications to the construction and classification of reflection groups [7]. Reflexive graphs that have been investigated so far are trees [4], [6], some classes of bicyclic graphs [10], [13] (see also [8]) and various classes of cactuses with more than two cycles [5], [9], [11], [12].…”