Independence numbers of some double vertex graphs and pair graphs

The combinatorial properties of double vertex graphs has been widely studied since the 90's. However only very few results are know about the independence number of such graphs. In this paper we obtain the independence numbers of the double vertex graphs of fan graphs and wheel graphs. Also we obtain the independence numbers of the pair graphs, that is a generalization of the double vertex graphs, of some families of graphs.Comment: 17 pages. Minor changes in the proof

The packing number of the double vertex graph of the path graph

Neil Sloane showed that the problem of determine the maximum size of a binary code of constant weight 2 that can correct a single adjacent transposition is equivalent to finding the packing number of a certain graph. In this paper we solve this open problem by finding the packing number of the double vertex graph (2-token graph) of a path graph. This double vertex graph is isomorphic to the Sloane's graph. Our solution implies a conjecture of Rob Pratt about the ordinary generating function of sequence A085680.Comment: 21 pages, 7 figures. V2: 22 pages, more figures added. V3. minor corrections based on referee's comments. One figure corrected. The title "On an error correcting code problem" has been change

Watershed Management to control Pollution in the Ayuquila River, Jalisco, Mexico

The Ayuquila River watershed is important to western Mexico because of its biodiversity, physiography, fisheries resources, and water production. However, human activities are continuingly affecting natural resources within the basin. Soil erosion, as result of land use change, agriculture in steep land, extensive grazing activities and forest fires; and water diversion and pollution of the Ayuquila River are two relevant issues that have affected the natural resources of this watershed. This river system plays an important role in wildlife conservation, containing 29 fish species, of which 12 are found inside the BRSM. The River also contains nine species of crustacean, one that is endemic to Jalisco State. The otter (Lontra longicaudis), a species threatened within the BRSM, is found in the Ayuquila- Armeria River watershed. This research focused on the reduction of river pollution and the reduction of negative impacts of water pollution delivered to those communities, some of the poorest in the state of Jalisco, that live downstream of the valley. This research was also designed as a way to increase the knowledge of soil erosion processes and water quantity and quality in tropical environments and to test and develop new tools that might facilitate parameter estimation and predictive capabilities within the Ayuquila River watershed. Research efforts in this dissertation had focused on the development of new scientific information about point and nonpoint-source pollution within the Ayuquila River based on three main research studies, the investigation into trail erosion, the production of an erosion sensitive map, and documenting and modeling water quantity and quality in the Ayuquila Watershed. Major concerns that this research seeks to contribute a solution, is to reduce negative impacts on public health, degradation in fisheries resources as source of local food supply, domestic water supplies for those communities, some of the poorest in the state of Jalisco, that live downstream the valley and that do not receive any benefit from the economic development of the Autlan- El Grullo Valley. Results from the commercial trail study, with sediment productions close to 100 ton/ha/yr, showed the importance of the application of conservation practices to reduce the potential erosion from commercial trails in my study area and potentially other tropical forests of Latin-American. The WEPP model used to predict soil erosion in the tropical mountain environments of Mexico was shown to be an adequate tool even with WEPP\u27s limitations for tropical soil environments. WEPP effectively contributed to the estimation of sediment plume production on trails, detected vegetation type differences in runoff and soil erosion, predicted the amount of rainfall as runoff well, and adequately developed soil erosion sensitive maps. Water diversion and pollution within the Ayuquila River are important sources of disturbance in the ecological conditions of riparian ecosystems. These two impacts cause a potential break in the ecological continuity of the Ayuquila River. Water quantity and quality modeling will provide opportunities for discussion and analysis of alternatives to water management and possible impacts to the river