Vertex-irregular Labeling and Vertex-irregular Total Labeling on Caterpillar Graph

For a simple graph G with the vertex set V and the edge set E, a labeling ?? : E(G) ?????? {1, 2, ?? ?? ?? , k} is called a vertex-irregular k-labeling on G if for any two different vertices x\ud and y in V we have wt(x) ??= wt(y) where wt(x) =\ud ???\ud z???V ??(xz). The irregularity strength\ud of G, denoted by s(G), is the smallest positive integer k for which G has a vertex-irregular\ud k???labelling. A labeling ?? : V (G) ??? E(G) ?????? {1, 2, ?? ?? ?? , k} is called a vertex-irregular total\ud k-labeling of G if for any two different vertices x and y in V we have wt(x) =?? wt(y) where\ud ???\ud z???V ??(xz). The total vertex irregularity strength of G, denoted by tvs(G), is the smallest positive integer k for which G has a vertex-irregular total k???labelling. In this paper, we determined the irregularity strength and the total vertex irregularity strength of a caterpillar graph