Abstract

In this paper we find the radio D-distance number of some standard graphs. If u, v are vertices of a connected graph G, the D-length of a connected u-v path s is defined as l^D(s) = l(s) + deg (v) + deg (u) + Σ deg(w), where the sum runs over all intermediate vertices w of s and l(s) is the length of the path. The D-distance d^D(u, v) between two vertices u, v of a connected graph G is defined a dᴰ(u, v) = min{l^D(s)}, where the minimum is taken over all u-v paths s in G. In other words, dᴰ(u, v) = min{l(s) + deg(v) + deg(u) + Σdeg(w)}, where the sum runs over all intermediate vertices w in s and minimum is taken over all u-v paths s in G. Radio D-distance coloring is a function ƒ : V(G) → N such that d^D(u, v) + |f(u)-f(v)| ≥ 〖diam〗^D(G) + 1, where 〖diam〗^D(G) is the D-distance diameter of G. A D-distance radio coloring number of G is the maximum color assigned to any vertex of G. It is denoted by 〖rn〗^D(G).