Group connectivity in line graphs

Tutte introduced the theory of nowhere zero flows and showed that a plane graph G has a face k-coloring if and only if G has a nowhere zero A-flow, for any Abelian group A with |A|≥k. In 1992, Jaeger et al. [9] extended nowhere zero flows to group connectivity of graphs: given an orientation D of a graph G, if for any b:V(G)?A with ∑_v∈V(G)b(v)=0, there always exists a map f:E(G)?A−{0}, such that at each v∈V(G), in A, then G is A-connected. Let Z₃ denote the cyclic group of order 3. In [9], Jaeger et al. (1992) conjectured that every 5-edge-connected graph is Z₃-connected. In this paper, we proved the following.

• (i) 

  Every 5-edge-connected graph is Z3-connected if and only if every 5-edge-connected line graph is Z3-connected.

• (ii) 

  Every 6-edge-connected triangular line graph is Z3-connected.

• (iii) 

  Every 7-edge-connected triangular claw-free graph is Z3-connected.

In particular, every 6-edge-connected triangular line graph and every 7-edge-connected triangular claw-free graph have a nowhere zero 3-flow.     

Adhesion and components of solid surface energies

Contact angle data for sets of probe liquids allow the determination of components of solid surface energies which in turn can be used to calculate the work of adhesion of other materials to the solid surface. There is much debate currently about the correct choice of the acid–base components for the probe liquids. For many systems, the strength of adhesion measured independently correlates well with the calculated work of adhesion. Recent trends in this area include adhesion under water and the adhesion of bacterial and other cells to immersed solids.     

Extremal hexagonal chains concerning largest eigenvalue

In this paper, we define a roll-attaching operation of a hexagonal chain, and prove Gutman’s conjecture affirmatively by using the operation. The idea of the proof is also applicable to the results concerning extremal hexagonal chains for the Hosoya index and Merrifield-Simmons index     

A superconvergence result of the natural convection equations

This paper presents a superconvergence result of the stationary natural convection equations. The superconvergence result is obtained by applying a coarse mesh projection for finite element approximation. This projection method is actually a postprocessing procedure that constructs a new approximation based on a high order polynomial on coarse mesh, under some regularity assumption. Finally, numerical experiment is presented for testing, which confirms the theoretic analysis. Copyright © 2015 John Wiley & Sons, Ltd.     

Gröbner-Shirshov basis for degenerate Ringel-Hall algebras of type <Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>

In this paper, by using the Frobenius morphism and the multiplication formulas of the generic extension monoid algebra, the authors first give a presentation of the degenerate Ringel-Hall algebra, and then construct the Gröbner-Shirshov basis for degenerate Ringel-Hall algebras of type F ₄.