Novel classes of integers and their applications in graph labeling

Adding new classes of integers to literature is both challenging and charming. Until a new class is completely characterized, mathematics is never going to be worth it. While it's absurd to play with integers without intended consequences. In this work, we introduce and investigate four new classes of integers namely, anti-totient numbers, half anti-totient numbers, near Zumkeller numbers and half near Zumkeller numbers by using the notion of non-coprime residues of $n$ including $n$. We formulate and propose relations of these new classes of numbers with previous well-known numbers such as perfect, totient, triangular, pentagonal, and hexagonal numbers. These new classes of integers have been completely characterized. Finally, as an application of these new classes of numbers, a new graph labeling is also proposed on anti-totient numbers.