A theorem on integer flows on cartesian products of graphs

It is shown that the Cartesian product of two nontrivial connected graphs admits a nowhere‐zero 4‐flow. If both factors are bipartite, then the product admits a nowhere‐zero 3‐flow. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 93–98, 2003     

Configuration and E/Z interconversion mechanism of O(S)-allyl-S(O)-methyl-N-(acridin-9-yl) iminothiocarbonate

The configuration and dynamic behavior of O-allyl-S-methyl-N-(acridin-9-yl)iminothiocarbonate (1) and its S-allyl-O-methyl regioisomer (2) were studied using quantum chemical calculations and by applying a novel graphical method to scatter maps obtained from MD simulations for evaluation of an NOE-weighted internuclear distance (r(NOE)). Energy calculations indicated that the Z configuration was predominant for each compound and, further, this was supported both by the calculated chemical shifts and the r(NOE). Both N-inversion- and rotation-type transition-state structures were also calculated for the E/Z isomerization process, the results indicating that the preferred interconversion mechanism for 1 is N-inversion, but contrastingly, interconversion via rotation is equally as probable as N-inversion for 2. This supports the notion that one or the other or both pathways can be active and each system needs to be assessed on a case-by-case basis.     

Regiospecific synthesis,structure and electron ionization mass spectra of 1,3‐thiazolidin‐4‐ones containing the acridine skeleton

The synthesis of regioisomeric 3‐alkyl(aryl)‐2‐(acridin‐9′‐yl)imino‐1,3‐thiazolidin‐4‐ones ( 8b‐i ) and 2‐alkyl(aryl)imino‐3‐(acridin‐9′‐yl)‐1,3‐thiazolidin‐4‐ones ( 11a‐i ) was performed by the reaction of 3‐(acridin‐9‐yl)‐1‐alkyl(aryl)thioureas 5a‐i with methyl bromoacetate and bromoacetyl bromide, respectively, via the corresponding isothiourea hydrobromides with excellent regioselectivity. The structure, NMR spectra and mass spectrometric behavior of the resulting compounds are discussed.     

On optimal embeddings of metrics in graphs

This paper extends previous results of the authors. In particular, non-treerealizable metrics are investigated and it is shown that every finite metric has an optimal realization by a graph.     

Explicit construction of regular graphs without small cycles

For every integerd>2 we give an explicit construction of infinitely many Cayley graphsX of degreed withn(X) vertices and girth >0.4801...(logn(X))/log (d−1)−2. This improves a result of Margulis. Dedicated to Paul Erdős on his seventieth birthday     

Associative products of graphs

It is shown that there exist exactly twenty products of graphs defined on the Cartesian product of the vertex sets of the factors where the adjacencies of the vertices only depend on the adjacencies in the factors.     

On graphs with regular groups

A new method for the construction of graphs with given regular group is developed and used to show that to every nonabelian group G of order 3^k, k ≥ 4, there exists a graph X whose automorphism group is isomorphic to G and regular as a permutation group on the set of vertices of X.     

On groups of polyhedral graphs

It is shown that every group of sense preserving automorphisms of a polyhedral graph is isomorphic to a group of rotations of the sphere.