Fault-tolerant Designs in Lattice Networks on the Klein Bottle

In this note, we consider triangular, square and hexagonal lattices on the flat Klein bottle, and find subgraphs with the property that for any $j$ vertices there exists a longest path (cycle) avoiding all of them. This completes work previously done in other lattices

On H-irregularity Strengths of G-amalgamation of Graphs

A simple graph G=(V(G),E(G)) admits an H-covering if every edge in E(G) belongs at least to one subgraph of G isomorphic to a given graph H. Then the graph G admitting H-covering admits an H-irregular total k-labeling f: V(G) U E(G) \to {1, 2, ..., k} if for every two different subgraphs H\u27 and H\u27\u27 isomorphic to H there is $wt_{f}(H\u27) \neq wt_{f}(H\u27\u27)$, where $wt_{f}(H)= \sum \limits_{v\in V(H)} f(v) + \sum \limits_{e \in E(H)} f(e)$ is the associated H-weight. The minimum k for which the graph G has an H-irregular total k-labeling is called the total H-irregularity strength of the graph G.In this paper, we obtain the precise value of the total H-irregularity strength of G-amalgamation of graphs

River Tea Tree Oil: Composition, Antimicrobial and Antioxidant Activities, and Potential Applications in Agriculture

Melaleuca is one of the genera of the Myrtaceae family enriched in tea tree oil (TTO). Tea tree oils of Melaleuca bracteata and Melaleuca alternifolia are of prime importance and have antioxidant and antimicrobial properties. Terpinen-4-ol and 1-8 cineole are major constituents of M. alternifolia oil. The percentages of the compounds in the oils can slightly vary according to the region of plant harvest, the distillation technique, or the part of the plant used for oil extraction. TTO has a bactericidal effect against various bacterial species such as Bacillus cereus, B. subtilis, E. coli, Pseudomonas putida, and S. aureus. Several reports proved that this essential oil is also effective against fungal strains of Fusarium, Aspergillus, and Candida species. It also has antioxidant properties such as radical scavenging activity and reducing power. The antioxidant properties of TTO at a concentration of 30 mM were observed to be greater than those of butylated hydroxytoluene (BHT), commonly used as a synthetic antioxidant. TTO is also an effective organic fungicide, herbicide, and insecticide for use in the agriculture sector. Postharvest application of the oil has been found efficient on sweet basil, citrus, and strawberry. It is concluded that tea tree oil has the potential to be used in the food, agriculture, and pharmaceutical industries as a natural antimicrobial and preservative agent. This review provides comprehensive information regarding the antioxidant and antimicrobial activities of tea tree oil and its potential applications in agriculture