The spectrum of the vertex quadrangulation of a 4-regular toroidal graph and beyond

Eigenvalue −1 of the Vertex Quadrangulation of a 4-Regular Graph

“…Next, take a cycle C 4 (quadrangle) and denote one pair of its diagonal vertices by numbers 1 and 2, respectively. A garland G p [24] in our text is a cyclic construct formed from p cycles C 4 (quadrangles), where vertex 1 of each copy of C 4 is connected by an edge to vertex 2 of an adjacent copy of C 4 , and conversely. Thus, each quadrangle in G p has two diagonal vertices of valency 3 and two other diagonal vertices that retain valency 2 and are not used in the construction for contacts.…”

“…Of course, speaking about the role of this eigenvalue, we are not disputing the role of any other values, but we will be mainly interested in the eigenvalue −1. In addition, we note that there are other "not our" studied cases [21][22][23][24][25][26][27], when this eigenvalue provides some structural information about the molecules whose graphs have it.…”

“…There are several possible topics that, in a broader context, could lead to the QG type of graphs indicated in the title of this text. One such topic, already sufficiently described in our previous work [24], is the spectra of graphs with decorated vertices. In our particular case, this means quadrangulating the vertices of the original 4-regular graph, or, in a simple loose definition, replacing each of its vertices, depicted as a point, with a quadrilateral and attaching connecting edges to the corners of adjacent quadrilaterals.…”

“…In the case of bipartite vertex-quadrangulated graphs, this becomes the inclusion of the full spectrum of the skeletal graph of the cube-{3, 1 3 , −1 3 , −3}-where each superscript means the multiplicity of the corresponding eigenvalue. Moreover, of particular interest is the subcase when there is not just inclusion of the spectrum of K 4 , but also a divisor isomorphic to this graph (which in the general case is not always realized) [24].…”

“…Speaking of chemical applications, we can recall the synthetic aspects of chemistry, where QG graphs, and therefore any useful information about them, can play a role. One such example has been discussed previously in [24], the desired synthesis of a twodimensional polymer [C 4 H 4 ] p of cyclobutadiene, C 4 H 4 , which has not yet been realized. If it comes true, then in the future it will also become possible to synthesize the same polymer directly using cheaper and more readily available acetylene C 2 H 2 , from which cyclobutadiene is obtained.…”