Shayou Permutation Labelling for Some Families of Graphs

This paper introduces a new graph labeling called Shayou Permutation Labeling. A formula is proposed to define the edge labels from the vertex labels. A function π is said to be a Shayou Permutation Labeling of a graph G if there exists a mapping from the vertex set of G to the set {1 , 2 ,…, p } such that each edge uv is assigned the label ࣊)࢛࢜= ( !൯)࢜)࣊ ା) ࢛)࣊൫ !(૚ି (࢜)࣊ ା) ࢛)࣊ ) ,and the resulting edge labels are distinct numbers. The proposed labeling is based on the additive relationship between the labels of the end vertices of each edge. This labeling provides a systematic approach for assigning edge labels using factorial expressions derived from vertex labels. The concept contributes to the study of graph labeling by introducing a new labeling technique that can be extended to other classes of graphs. The existence of this labeling is studied for some classes of graphs, namely, the path graph and star graph. The results demonstrate that these graph families admit Shayou Permutation Labeling under the proposal labeling rule. Graph labelling can be classified into different types which is cited as [9,13,14,15,16,17]. Graph labeling concept can be extended to automata theory which is cited as [10, 11, 12]. Graph labeling is also extended to domination [1, 6, 7, 8].