On The Total Irregularity Strength of Regular Graphs

Let ðº = (ð‘‰, ð¸) be a graph. A total labeling ð‘“: 𑉠∪ ð¸ → {1, 2, ⋯ , ð‘˜} iscalled a totally irregular total ð‘˜-labeling of ðº if every two distinct vertices ð‘¥ and𑦠in 𑉠satisfy ð‘¤ð‘“(ð‘¥) ≠ð‘¤ð‘“(ð‘¦) and every two distinct edges ð‘¥1ð‘¥2 and ð‘¦1ð‘¦2 in ð¸satisfy ð‘¤ð‘“(ð‘¥1ð‘¥2) ≠ð‘¤ð‘“(ð‘¦1ð‘¦2), where ð‘¤ð‘“(ð‘¥) = ð‘“(ð‘¥) + Σð‘¥ð‘§âˆˆð¸(ðº) ð‘“(ð‘¥ð‘§) andð‘¤ð‘“(ð‘¥1ð‘¥2) = ð‘“(ð‘¥1) + ð‘“(ð‘¥1ð‘¥2) + ð‘“(ð‘¥2). The minimum 𑘠for which a graph ðº hasa totally irregular total ð‘˜-labeling is called the total irregularity strength of ðº,denoted by ð‘¡ð‘ (ðº). In this paper, we consider an upper bound on the totalirregularity strength of ð‘š copies of a regular graph. Besides that, we give a dual labeling of a totally irregular total ð‘˜-labeling of a regular graph and we consider the total irregularity strength of ð‘š copies of a path on two vertices, ð‘š copies of a cycle, and ð‘š copies of a prism ð¶ð‘› â–¡ ð‘ƒ2