Total Distance Vertex Irregularity Strength of Hairy Cycle C_m^n Graph

A total graph labeling is an assignment of integers to the union of vertices and edges to certain conditions. The labeling becomes D -distance vertex irregular total k-labeling when each vertex of G has a different weight (which is determined by D-distance neighborhood). The total distance vertex irregularity strength of G denoted by tdis(G) and define as the minimum of the biggest label k over all D-distance vertex irregular total k-labelings of G. In this paper, we investigate about D-distance vertex irregular total k-labelings on hairy cycle C_m^n graphs which can be applied to cryptography and computational networks. The unique hairy cycle graph construction makes the weights of each vertex of this graph different and random. Therefore, this weight formula can be applied in stream cipher cryptography as a key generator. To obtained the formula labeling, we carried out labeling experiments repeatedly to find labeling patterns and then formulate it into a labeling function. We also provide the lower bound and determine the value of total distance vertex irregularity strength of hairy cycle C_m^n graphs. we prove that for m=2,3,4, n≥5 an odd positive integer , hairy cycle C_m^n graphs admits an D-distance vertex irregular total k-labelings with total distance vertex irregularity strength, tdis(C_m^n )=⌈(mn+1)/2⌉.