Hypercube embeddings and Cayley graphs generated by transpositions

A graph is called a partial cube if it can be embedded into a hypercube isometrically. In this paper, we study a class of Cayley graphs —Cayley graphs generated by transpositions and show that a Cayley graph Γ generated by transpositions is a partial cube if and only if Γ is a bubble sort graph. This result enhances a result of Alahmadi et al. [Math. Meth. Appl. Sci. 39 (2016), 4856–4865]: BS is a partial cube. As a corrollary, we give the analytical expressions of the Wiener indices of bubble sort graphs.

Partition and Colored Distances in Graphs Induced to Subsets of Vertices and Some of Its Applications

If G is a graph and P is a partition of V(G), then the partition distance of G is the sum of the distances between all pairs of vertices that lie in the same part of P. A colored distance is the dual concept of the partition distance. These notions are motivated by a problem in the facility location network and applied to several well-known distance-based graph invariants. In this paper, we apply an extended cut method to induce the partition and color distances to some subsets of vertices which are not necessary a partition of V(G). Then, we define a two-dimensional weighted graph and an operator to prove that the induced partition and colored distances of a graph can be obtained from the weighted Wiener index of a two-dimensional weighted quotient graph induced by the transitive closure of the Djoković–Winkler relation as well as by any partition that is coarser. Finally, we utilize our main results to find some upper bounds for the modified Wiener index and the number of orbits of partial cube graphs under the action of automorphism group of graphs.

Covering Partial Cubes with Zones

A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of a hypercube in which the graph can be embedded. We consider the problem of covering the vertices of a partial cube with the minimum number of zones. The problem admits several special cases, among which are the following:cover the cells of a line arrangement with a minimum number of lines,select a smallest subset of edges in a graph such that for every acyclic orientation, there exists a selected edge that can be flipped without creating a cycle,find a smallest set of incomparable pairs of elements in a poset such that in every linear extension, at least one such pair is consecutive,find a minimum-size fibre in a bipartite poset.We give upper and lower bounds on the worst-case minimum size of a covering by zones in several of those cases. We also consider the computational complexity of those problems, and establish some hardness results.

Silsesquioxanes as molecular analogues of single-site heterogeneous catalysts

We discuss herein selected examples of metal complexes of polyhedral oligosilses-quioxanes (POSSs) as models of single-site heterogeneous surface catalysts. The utility of these compounds as such models is illustrated when employed as analogues of single-site titanium species supported on a silica surface. Deep insights into structure–functionality relationships can be gained. In particular, it was possible to probe the relationship between accessibility of the reactive centre and turnover frequencies in a manner that is impossible for a purely heterogeneous catalyst. We also report that the partially dehydroxylated SiO 2 surface alone is an effective radical polymerization initiation catalyst. This surface reactivity is modelled by the solution reactions between the olefin substrate and two POSSs, the completely condensed triganol prism, Si 6 Cy 6 O 9 ( a6b0 , Cy = Cyclohexyl, C 6 H 11 ), and the incompletely condensed partial cube, Si 7 Cy 7 O 9 (OH) 3 ( a7b3 ). The former, with six-membered Si 3 O 3 rings, is a catalyst. The latter, without this feature, is not. Similar reactivity discrimination is observed in the gas phase reactions of these POSSs with the olefin substrate, examined using atmospheric pressure chemical ionization- and collision-induced decomposition spectroscopies. Silsesquioxane a6b0 , containing Si 3 O 3 rings, reacts with the olefin, forming grafted olefin monomers and dimers, while this reactivity is not observed with silsesquioxane a7b3 .

Cubic Partial Cubes from Simplicial Arrangements

We show how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane. As a consequence, we find nine new infinite families of cubic partial cubes as well as many sporadic examples.