d-Lucky Labeling of Honeycomb Network - Sahayamary, Rini Angeline A; Arockiamary, Teresa S (International Journal of Computer Sciences and Engineering, 2025)
Let f:V(G)⟶N be a labeling of the vertices of a graph G by positive integers. Let S(v) denote the sum of labels of the neighbors of the vertex v in G. If v is an isolated vertex of G we put S(v)=0. A labeling f is lucky if S(u)≠S(v) for every pair of adjacent vertices u and v. The lucky number of a graph G, denoted by η(G), is the least positive integer k such that G has a lucky labeling with 1,2,…k as the set of labels. Let l:V(G)⟶ 1,2,…k be a labeling of the vertices of a graph G by positive integers. Define c(u)=d(u) + ∑_(v∈N(u))▒〖l(v)〗 where d(u) denotes the degree of u and N(u) denotes the neighbourhood of u. We define a labeling l as d-lucky if c(u)≠c(v), for every pair of adjacent vertices u and v in G. The d-lucky number of a graph G, denoted by η_dl (G), is the least positive integer k such that G has a d-lucky labeling with 1,2,…k as the set of labels. In this paper, we study d-lucky labeling of Honeycomb network and Honeycomb torus network. Further we have obtained the d-lucky number for Honeycomb network and Honeycomb torus network.