Total Edge Irregularity Strength of the Cartesian Product of Bipartite Graphs and Paths

Abstract

For a simple graph G = (V (G), E(G)), a total labeling ∂ is called an edge irregular total k-labeling of G if ∂ : V (G) ∪ E(G) → {1, 2, . . . , k} such that for any two different edges uv and u'v' in E(G), we have wt∂(uv) not equal to wt∂(u'v') where wt∂(uv) = ∂(u) + ∂(v) + ∂(uv). The minimum k for which G has an edge irregulartotal k-labeling is called the total edge irregularity strength, denoted by tes(G). It is known that ceil((|E(G)|+2)/3) is a lower bound for the total edge irregularity strength of a graph G. In this paper we prove that if G is a bipartite graph for which this bound is tight then this is also true for Cartesian product of G with any path.

Total Face Irregularity Strength of Certain Graphs

“…They discovered that for any graph, the irregularity strength sðGÞ: , which is the lowest possible value k for which G constitutes an irregular assignment with label at most k, is extremely difficult to find. In recent times, motivated by this particular concept, many researchers are having specific interest for these types of irregular labeling and have found the irregularity strength for some graphs [5][6][7][8][9][10]. Shabbir et al [11] have proved their exact values of the strength of total vertex (edge) irregularities of a randomly convex unions of (3,6)-fullerene graphs.…”