Problems on Matchings and Independent Sets of a Graph

Let $G$ be a finite simple graph. For $X \subset V(G)$, the difference of $X$, $d(X) := |X| - |N (X)|$ where $N(X)$ is the neighborhood of $X$ and $\max \, \{d(X):X\subset V(G)\}$ is called the critical difference of $G$. $X$ is called a critical set if $d(X)$ equals the critical difference and ker$(G)$ is the intersection of all critical sets. It is known that ker$(G)$ is an independent (vertex) set of $G$. diadem$(G)$ is the union of all critical independent sets. An independent set $S$ is an inclusion minimal set with $d(S) > 0$ if no proper subset of $S$ has positive difference. A graph $G$ is called K\"onig-Egerv\'ary if the sum of its independence number ($\alpha (G)$) and matching number ($\mu (G)$) equals $|V(G)|$. It is known that bipartite graphs are K\"onig-Egerv\'ary. In this paper, we study independent sets with positive difference for which every proper subset has a smaller difference and prove a result conjectured by Levit and Mandrescu in 2013. The conjecture states that for any graph, the number of inclusion minimal sets $S$ with $d(S) > 0$ is at least the critical difference of the graph. We also give a short proof of the inequality $|$ker$(G)| + |$diadem$(G)| \le 2\alpha (G)$ (proved by Short in 2016). A characterization of unicyclic non-K\"onig-Egerv\'ary graphs is also presented and a conjecture which states that for such a graph $G$, the critical difference equals $\alpha (G) - \mu (G)$, is proved. We also make an observation about ker$G)$ using Edmonds-Gallai Structure Theorem as a concluding remark.Comment: 18 pages, 2 figure

Carotid intima media thickness and low high-density lipoprotein (HDL) in South Asian immigrants: could dysfunctional HDL be the missing link?

IntroductionSouth Asian immigrants (SAIs) in the US exhibit higher prevalence of coronary artery disease (CAD) and its risk factors compared with other ethnic populations. Conventional CAD risk factors do not explain the excess CAD risk; therefore there is a need to identify other markers that can predict future risk of CAD in high-risk SAIs. The objective of the current study is to assess the presence of sub-clinical CAD using common carotid artery intima-media thickness (CCA-IMT), and its association with metabolic syndrome (MS) and pro-inflammatory/dysfunctional HDL (Dys-HDL).Material and methodsA community-based study was conducted on 130 first generation SAIs aged 35-65 years. Dys-HDL was determined using the HDL inflammatory index. Analysis was completed using logistic regression and Fisher's exact test.ResultsSub-clinical CAD using CCA-IMT ≥ 0.8 mm (as a surrogate marker) was seen in 31.46%. Age and gender adjusted CCA-IMT was significantly associated with type 2 diabetes (p = 0.008), hypertension (p = 0.012), high-sensitivity C-reactive protein (p < 0.001) and homocysteine (p = 0.051). Both the presence of MS and Dys-HDL was significantly correlated with CCA-IMT, even after age and gender adjustment. The odds of having Dys-HDL with CCA-IMT were 5 times (95% CI: 1.68, 10.78).ConclusionsThere is a need to explore and understand non-traditional CAD risk factors with a special focus on Dys-HDL, knowing that SAIs have low HDL levels. This information will not only help to stratify high-risk asymptomatic SAI groups, but will also be useful from a disease management point of view

Influence of temperature on the preservation of fish -Winter School on Impact of Climate Change on Indian Marine Fisheries held at CMFRI, Cochin 18.1.2008 to 7.2.2008

Live fish muscle is relaxed and elastic. Immediately after death rigor mortis sets in, then the whole body becomes inflexible and rigid. The onset of rigor depends upon the temperature, particularly on the difference of temperatures between that of water and storage. If the difference is great, shorter is the time from death to rigor. Fish being the most perishable of human foods, they start spoiling the moment they are taken out of water and die. Temperature being a very important factor accelerating the process of spoilage, in a tropical country like India the ambient temperatures are very conductive for causing quick spoilage in fish. The spoilage reactions commencing on the death of the fish proceed at a very rapid pace. The rate of spoilage of fish at 2.5°C is twice as fast as that at –1.1°C. At 5.5°C, it is twice and at 11°C it is four times as fast as that at 0°C. The spoilage cannot be stopped completely. The best that can be done is to slow it down by means of some refrigeration technique, the simplest of which is addition of ice

Image Segmentation Based on Doubly Truncated Generalized Laplace Mixture Model and K Means Clustering

The present paper aims at performance evaluation of Doubly Truncated Generalized Laplace Mixture Model and K-Means clustering (DTGLMM-K) for image analysis concerned to various practical applications like security, surveillance, medical diagnostics and other areas. Among the many algorithms designed and developed for image segmentation the dominance of Gaussian Mixture Model (GMM) has been predominant which has the major drawback of suiting to a particular kind of data. Therefore the present work aims at development of DTGLMM-K algorithm which can be suitable for wide variety of applications and data. Performance evaluation of the developed algorithm has been donethrough various measures like Probabilistic Rand index (PRI), Global Consistency Error (GCE) and Variation of Information (VOI). During the current work case studies forvarious different images having pixel intensities has been carried out and the obtained results indicate the superiority of the developed algorithm for improved image segmentation

Improved Hybrid Algorithm Approach based Load Balancing Technique in Cloud Computing

The routine life of modern citizens is completely dominated by the computer aided services The computer aided services depends on information and communication technologies The success behind this cloud computing are data centers with virtualization technology equipped with fastest internet and the wide acceptance of the users due to its affordable price to the common people Effective services can be provided to the end user only when proper scheduling of tasks are done in peak hours when heterogeneous collection of requests are coming to the data cente