Average distances in square‐cell configurations

A square‐cell configuration (“square animal”) is a subgraph of the square lattice in which all inner faces are 4‐cycles. We determine explicit expressions for the sum (W) of the (topological) distances between all pairs of vertices of a square‐cell configuration, as well as for the related average distance $\overline{W}$. Such expressions are deduced for several families of symmetric square‐cell configurations. For instance, if O(n) stands for the octagonal square‐cell configuration with n circular levels, then W(O(n))=(211/5)n⁵−(181/3)n⁴+(109/3)n³−(35/3)n²+(22/15)n and W (O(n))=2[(7n²−10n+4)(7n²−10n+3)]⁻¹W(O(n)). © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 611–617, 2000     

A characterization of 1-cycle resonant graphs among bipartite 2-connected plane graphs

It is proved that a bipartite 2-connected plane graph in which the common boundary of adjacent faces is a simple curve is 1-cycle resonant if and only if the outer face of $G$ is alternating and each inner vertex has degree two. This extends a result from [X. Guo, F. Zhang, $k$-cycle resonant graphs, Discrete Math. 135 (1994) 113–20] that a hexagonal system is 1-cycle resonant if and only if it is catacondensed.     

Vizing's conjecture: a survey and recent results

Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw‐free graphs with arbitrary graphs. Open problems, questions and related conjectures are discussed throughout the paper. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 46–76, 2012     

Investigation of corannulene for molecular hydrogen storage via computational chemistry and experimentation

Molecular simulations for hydrogen physisorption with corannulene molecules arranged according to their crystal structure result in good agreement with the weight-percent hydrogen stored as determined experimentally employing a 3-g sample of highly crystalline corannulene at ambient temperatures and 72 bar of pressure. Calculated enthalpies of adsorption for corannulene/hydrogen molecular systems obtained from ab initio calculations which take into account electron correlation via second-order M?ller-Plesset perturbation theory are in good agreement with literature experimental enthalpies of adsorption for activated carbons interacting with molecular hydrogen. Ab initio results also show that corannulene molecules arranged in a sandwich structure are important for approximately doubling the binding energy of corannulene interacting with molecular hydrogen through a cooperative interaction. To test the effects of finite temperatures and pressures, stack arrays were used as input for molecular dynamics simulations and indicate that physisorption mechanisms including van der Waals forces and dipole-induced dipole interactions may yield enhanced adsorption capacity in relation to other carbon-based materials. These results will be instrumental in identifying interlayer separations of an array of corannulene or related molecules that may provide a high weight percent of physisorbed hydrogen.     

Engineering process as a focus: STEM professional development with elementary STEM‐focused professional development schools

Young children are capable of engaging in STEM investigations when they are guided by skilled and knowledgeable teachers. However, many elementary teachers may lack sufficient STEM content knowledge and report feeling unprepared to teach STEM content. Two university faculty members in mathematics and science education, worked to cultivate and advance two designated Elementary STEM‐Focused professional development schools through a two year series of an after‐school STEM professional development (PD) Program. As the STEM PD Program progressed, it became evident that teachers were interested in and needed more experiences with the elements of the engineering process for young learners. With this in mind, several of the PD sessions were designed to highlight the engineering process and allow teachers to experience various activities that would engage young learners. To examine how this focus on the engineering process impacted the teachers in this STEM PD Program, a research study was organized during year two of the STEM PD Program. The results of this study provide evidence that this program had a positive influence on the teacher participants’ engineering teacher efficacy and implementation of engineering lessons and activities within their classrooms.     

Two-ended regular median graphs

We show that regular median graphs of linear growth are the Cartesian product of finite hypercubes with the two-way infinite path. Such graphs are Cayley graphs and have only two ends.For cubic median graphs G the condition of linear growth can be weakened to the condition that G has two ends. For higher degree the relaxation to two-ended graphs is not possible, which we demonstrate by an example of a median graph of degree four that has two ends, but nonlinear growth.     

Coloring graph bundles

Graph bundles generalize the notion of covering graphs and products of graphs. Several results about the chromatic numbers of graph bundles based on the Cartesian product, the strong product and the tensor product are presented. © 1995 John Wiley & Sons, Inc.     

Asymptotic Properties of Fibonacci Cubes and Lucas Cubes

It is proved that the asymptotic average eccentricity and the asymptotic average degree of both Fibonacci cubes and Lucas cubes are \({(5+\sqrt{5})/10}\) and \({(5-\sqrt{5}) /5}\) , respectively. A new labeling of the leaves of Fibonacci trees is introduced and it is proved that the eccentricity of a vertex of a given Fibonacci cube is equal to the depth of the associated leaf in the corresponding Fibonacci tree. Hypercube density is also introduced and studied. The hypercube density of both Fibonacci cubes and Lucas cubes is shown to be \({(1-1/\sqrt{5})/ \rm log_ {2}\varphi}\) , where \({\varphi}\) is the golden ratio, and the Cartesian product of graphs is used to construct families of graphs with a fixed, non-zero hypercube density. It is also proved that the average ratio of the numbers of Fibonacci strings with a 0 (a 1, respectively) in a given position, where the average is taken over all positions, converges to \({\varphi^{2}}\) , and likewise for Lucas strings.