Total edge irregularity strength of complete graphs and complete bipartite graphs

AbstractA total edge irregular k-labelling ν of a graph G is a labelling of the vertices and edges of G with labels from the set {1,…,k} in such a way that for any two different edges e and f their weights φ(f) and φ(e) are distinct. Here, the weight of an edge g=uv is φ(g)=ν(g)+ν(u)+ν(v), i. e. the sum of the label of g and the labels of vertices u and v. The minimum k for which the graph G has an edge irregular total k-labelling is called the total edge irregularity strength of G.We have determined the exact value of the total edge irregularity strength of complete graphs and complete bipartite graphs

Looseness of Planar Graphs

International audienceA face of a vertex coloured plane graph is called loose if the number of colours used on its vertices is at least three. The looseness of a plane graph G is the minimum k such that any surjective k-colouring involves a loose face. In this paper we prove that the looseness of a connected plane graph G equals the maximum number of vertex disjoint cycles in the dual graph G increased by 2. We also show upper bounds on the looseness of graphs based on the number of vertices, the edge connectivity, and the girth of the dual graphs. These bounds improve the result of Negami for the looseness of plane triangulations. We also present infinite classes of graphs where the equalities are attained

Fullerene graphs have exponentially many perfect matchings

A fullerene graph is a planar cubic 3-connected graph with only pentagonal and hexagonal faces. We show that fullerene graphs have exponentially many perfect matchings.Comment: 7 pages, 3 figure

On universal graphs for hom-properties

A graph property is any isomorphism closed class of simple graphs. For a simple finite graph H, let → H denote the class of all simple countable graphs that admit homomorphisms to H, such classes of graphs are called hom-properties. Given a graph property , a graph G ∈ is universal in if each member of is isomorphic to an induced subgraph of G. In particular, we consider universal graphs in → H and we give a new proof of the existence of a universal graph in → H, for any finite graph H

Size Effect in Plastic Deformation and Failure of Metallic Glasses

Depending on the composition and structure of metallic glasses cells with the dimensions in the range from tenths nanometers to tenths micrometers were observed on the ductile fracture surface. The variation in dimple size was compared with the serrations presented on the loading curve at the nanoindentation of the metallic glasses with different compositions. Higher instantaneous deformation can be connected with simultaneous shearing at more suitable shear band configurations. The cell morphology with the various cell sizes is observed at the failure of the metallic glasses. At the failure of high strength metallic glasses, the cells are formed in short time due to the release of high amount of stored elastic energy. In this case the uniform cell morphology with the cell size of about 20 nm is observed

Crack propagation in metallic glass ribbon as a function of the position of stress concentrators

An amorphous metallic ribbon of Fe40Ni40B20 was used for in-situ observation of the crack propagation and shear band formation during tensile tests. Prior to the tensile tests, two holes (with different positions with respect to the tensile axis) were made by laser ablation as stress concentrators. The nucleation and propagation of shear bands on the ribbon surface during tensile tests were analysed with scanning electron microscopy (SEM). At room temperature inhomogeneous plastic deformation of amorphous alloy occurs via the development of primary and secondary shear bands. The influence of the different loading geometry on the topology of shear bands and crack propagation was studied