On some families of Linear Diophantine graphs

Abstract

Linear Diophantine labeling of graphs is an extension of the prime labeling of graphs. In this manuscript, we introduce some necessary conditions for determining whether a given graph admits Linear Diophantine labeling or not, and if yes, we will find such a Linear Diophantine labeling. We also study specific families of graphs, including the Complete graphs Kn; Wheel graphs Wn and Wn,n; Circulant graphs Cn(j); Path graphs Pn(j); Cartesian product graphs C3 × Cm; Normal Product graphs Pn ◦ Pn; Corona graphs G ⊙ H; Double Fan graphs gn = Pn + K2; Power graphs P2n and P3n, to ascertain their Linear Diophantine nature. We refer to a Linear Diophantine Graph as an LDG.