Gaussian Tribonacci R-Graceful Labeling of Some Tree Related Graphs

Let r be any natural number. An injective function , where  is the Gaussian Tribonacci number in the Gaussian Tribonacci sequence is said to be Gaussian Tribonacci r-graceful labeling if the induced edge labeling such that  is bijective. If a graph G admits Gaussian Tribonacci r-graceful labeling, then G is called a Gaussian Tribonacci r-graceful graph. A graph G is said to be Gaussian Tribonacci arbitrarily graceful if it is Gaussian Tribonacci r-graceful for all r. In this paper we investigate the Path graph , the Comb graph , the Coconut tree graph the regular caterpillar graph , the Bistar graph  and the Subdivision of Bistar graph are Gaussian Tribonacci arbitrarily graceful

Effect of vitamin D3 in treating hyperthyroidism in patients with graves’ disease

Background Graves’ disease (GD) is an autoimmune disease characterized by hyperthyroidism secondary to circulating autoantibodies. Multiple factors contributed to its etiology, including genetic and environmental factors. The role of vitamin D is well-known in calcium metabolism and skeletal homeostasis. Vitamin D was shown to be a modulator in both innate and adaptive immunity. There is a link between vitamin D deficiency and various autoimmune diseases. The prevalence of vitamin D deficiency was reported to be common in patients with GD. Interestingly, vitamin D deficiency is found to be associated with higher thyroid volume in patients with newly-onset GD. However, vitamin D deficiency relationship with GD remains a controversial issue. Objective The objective of this study was to evaluate the effect of vitamin D supplementation in GD with and without ophthalmopathy. Patients and methods A randomized prospective study was conducted on 60 adult patients with GD aged 20–40 years. Group 1 comprised 20 patients with GD receiving a daily dose of 30 mg of methimazole alone. Group 2 comprised 40 patients with GD receiving the same dose of methimazole, supplemented with intramuscular injection of vitamin D3 200 000 IU/month for 3 months. Patients were followed up over a 3-month duration. Results There was hypovitaminosis D in all participants with a percentage of vitamin D deficiency (vitamin D level: <20 ng/ml) of 73.9% in male and 54.1% in female and a vitamin D insufficiency (vitamin D level: 20–29 ng/ml) of 26.1% in male and 45.9% in female. Vitamin D was significantly correlated with thyroid volume and degree of exophthalmos. On vitamin D supplementation, group 2 had significantly lower thyroid volume and better effect on the degree of exophthalmos. Conclusion Vitamin D supplementation for GD has a favorable effect on thyroid volume and on the degree of exophthalmos