Prime cordial labeling of some special graphs

Some graph types, such as m-polygonal snakes and certain instances of brush graphs, are specified in this paper as primitive cordial labelling. For a bijection ƒ from the set of graph vertex sets V to the set of integers from 1 to the graph’s order to be considered a prime cordial labelling, there are precise conditions that each edge uv’s labelling must meet. If the label of vertex v is greater than 1, then the label of edge uv is 1, and if it is not, then the label of the edge is 0. As an added restriction, the ratio of edges indicated as 0 to those marked as 1 must not exceed 1:1. Two important results are presented in this study: Snake Identification for the Primal Mind: Prime cordial labelling can be shown to work for m-polygonal snakes, where m is equal to 10 plus (2n - 2) times 3, and n ranges from 1 to n. A unique labelling scheme that satisfies the prime cordiality requirements is used by the structures, which are composed of a series of polygons connected at their edges. It can be demonstrated that the brush graph Bn, where n≥14 and n≡2 (mod 3) for even n, can have prime pleasantly labelled graphs by exchanging the vertex un. By adhering star graphs to each vertex of a path graph, complex structures can be observed in brush graphs, which nonetheless meet the prime friendly labelling condition. By finding the best friendly labelling for various graph topologies, we contribute to graph theory and explain the properties of these complex networks. These kinds of findings also make it easier to study graph labelling and all the things it could be used for in the future.