# Prime Distance Graphs and Applications of Graph Labeling: A Modelling Approach

A diagram H is a prime distance graph if there is a one-to-one function h: V (H) Z such that the number |h(x) h(y)| is a prime for any two neighbouring vertices x and y. As a result, H is a prime distance graph if and only if it has a prime distance labelling. If H’s edge labels are distinct as well, h becomes a distinct prime distance labelling of H, and H is referred to as a distinct prime distance graph. The generalised Petersen graphs P (n, k) are defined as a graph with 2n (n 3) vertices and subscripts modulo n, with V (P (n, k)) = vi, ui: 0 I n 1 and E(P (n, k)) = vi vi+1, vi ui, ui ui+k: 0 I n 1. We highlight that the extended Petersen graphs P (n, 3) allowed a prime distance labelling for all even n > 5, and we postulate that P (n, 2) and P (n, 3) admit a prime distance labelling for any n 5 and all odd n 5, respectively, in this chapter. Furthermore, for all n 3, the cycle Cn admits a separate prime distance labelling.

**Author(s) Details:**

**Ajaz Ahmad Pir,
**Department of Mathematics, School of Chemical Engineering and Physical Sciences, Lovely Professional University Phagwara-144 411, Punjab, India.

**A. Parthiban,**

Department of Mathematics, School of Chemical Engineering and Physical Sciences, Lovely Professional University Phagwara-144 411, Punjab, India.

**Please see the link here: ****https://stm.bookpi.org/RAMRCS-V10/article/view/6273**