Complete bipartite graph is a totally irregular total graph

Abstract

A graph G is called a totally irregular total k-graph if it has a totally irregular total k-labeling λ : V ∪ E → {1, 2, • • • , k}, that is a total labeling such that for any pair of different vertices x and y of G, their weights wt(x) and wt(y) are distinct, and for any pair of different edges e and f of G, their weights wt(e) and wt(f ) are distinct. The minimum value k under labeling λ is called the total irregularity strength of G, denoted by ts(G). For special cases of a complete bipartite graph K m,n , the ts(K 1,n ) and the ts(K n,n ) are already determined for any positive integer n. Completing the results, this paper deals with the total irregularity strength of complete bipartite graph K m,n for any positive integer m and n.

The total irregularity strength of m copies of the friendship graph

“…They [8] proved that the lower bound is sharp for path graph of order 𝑛 ≠ 2, 5 and cycle graphs. Ramdani and Salman [9], Ramdani, et al [10], Tilukay, et al [11][12][13], also confirmed the sharpness of the lower bound of the total irregularity of several types of graphs, including friendship graph 𝐹 𝑛 . More results can be seen in a survey provided by Galian in [14].…”

Total Edge and Vertex Irregular Strength of Twitter Network

“…On interconnected networks such as the Beneš network and the Butterfly network, the test scores have been found by Rajasingh.I [8]. Nurdin [9]- [12] determines the tes, tvs,and ts of the second level Butterfly M [15] found a total irregular strength on a Complete Bipartite graph. Nandini, G. Kirithiga, et al [16] discussed the COVID-19 pandemic tree model using a complete k-ary tree network topology, Cayley tree network, Christmas trees, and the corona product of Christmas tree.…”