Abstract In this paper we define some new labellings for trees, called the in-improper and out-improper odd-graceful labellings such that some trees labelled with the new labellings can induce graceful graphs having at least a cycle. We, next, apply the new labellings to construct large scale of graphs having improper graceful/odd-graceful labellings or having graceful/odd-graceful labellings.

This is a complement to the comparison theorem for improper integrals in the textbook. The vanilla version presented in the textbook is good enough to solve some very easy examples and it becomes exponentially gory with the complexity of the integral. Fortunately it is not hard to refine the statement in the book, and turn it into a powerful tool to estimate the convergence of arbitrarily compl...

Diabetes is a major public health problem allover the world. Compliance refers to a patient both agreeing to and then undergoing some part of their treatment program as advised by their doctor or other health care workers. Aim: This study was conducted to demonstrate the pattern of compliance among diabetic patients, its determinants and reasons beyond noncompliance. Methodology: This cross sec...