Graceful Antimagic Path and Cycle Related Graphs

A graph with vertices and edges is said to be antimagic if its edges can be labeled with such that the weights of vertices of are pairwise distinct. The graceful labeling of a graph with edges is an assignment of integers from the set to the vertices of , such that no two vertices receive the same label, where each edge is assigned the absolute value of the difference between the labels of its end vertices and the resulting edge labeling runs from to is inclusive. Moreover, if the induced edge labeling is simultaneously antimagic, that is, the sums of the labels of all edges incident to a given vertex are pairwise distinct for different vertices, we call the graceful labeling graceful antimagic. In this study, we will exhibit the existence of graceful antimagic labeling for two families of graphs the first derived from the path graph Pn, and the second from the cycle graph Cn. Both families were derived using the idea of a rooted product between the two graphs.