On total edge and total vertex irregularity strength of pentagon cactus chain graph with pendant vertices

Let G ( V, E ) be a graph with a non-empty set of vertices V and a set of edges E . A total k -labeling f: V ∪ E → {1,2, …, k} is called an edge irregular total k -labeling if the weight of each edge is distinct, where the weight of an edge e = xy is wt(e) = f(xy) + f(x) + f(y). Whereas, f is a vertex irregular total k -labeling if the weight wt ( x ) ≠ wt ( y ) for two distinct vertices x , y of V ( G ) where the weight wt ( x ) = f ( x ) + ∑ υx∈E ( G ) f ( υx ). The minimum k for which G has an edge irregular total k -labeling is called the total edge irregularity strength (tes) of G . Further, the total vertex irregularity strength (tvs) of G is the minimum k for which G has a vertex irregular total k -labeling. In this paper, we find tes and tvs of heptagon cactus chain graph with pendants and get the results as follows: tes ( C ( P t r 3 ) ) = ⌈ 8 r + 2 3 ⌉ and tvs ( C ( P t r 3 ) ) = ⌈ 6 r + 3 4 ⌉ .