Non-Inclusive and Inclusive Distance Irregularity Strength of Complement and Split Graphs - Maryati, Tita Khalis; Fakhrurrozi, Fawwaz (Research Center of Inorganic Materials and Coordination Complexes, FMIPA Universitas Sriwijaya, 2026)
Let f be a map from vertices of a graph G to number from 1 to k. The labeling f is called distance irregular if for every two vertices x and y, it holds that wt_f(u) wt_f(v) where a weight wt_f(u) is defined as the sum of labels of the neighbors of u. Moreover, the labeling f is called inclusive distance irregular if for every two vertices x and y, wt_f(u) wt_f(v) with a weight wt_f(u) is defined as the sum of the label of u and the labels of the neighbors of u. The least number k where there exists a distance irregular labeling (resp. inclusive distance irregular labeling) is called a distance irregularity strength (inclusive distance irregularity strength), denoted by dis(G) (dis(G)). In this paper, we present a connection of distance irregular labeling and inclusive distance irregular labeling in a graph with its complement. In particular, we derive a new upper bound for distance irregularity strength and inclusive distance irregularity strength of any graph. Further, we determine the dis(G) and dis(G) for certain special family of split graph G and provide examples of a graph G satisfying dis(G) = dis(G).