EXPLORATION OF DECAGONAL GRACEFUL LABELING IN PATH GRAPHS WITH MUSICAL APPLICATIONS

Abstract

This study introduces a novel labeling technique called decagonal graceful labeling. Assume graph  is simple and finite, that has  edges and  vertices. The decagonal number denoted as  corresponds to the  decagonal number and a decagonal graceful labeling is defined by an injective function  that assigns the vertices in set  to the set . A bijective function  is naturally induced on the edge set by this vertex labeling, which assigns labels from the set  so that  for each edge
. A graph that allows this type of labeling is termed a decagonal graceful graph. We provide a technique for producing decagonal graceful labeling for a specific group of path-related graphs and show how this labeling may be extended to graphs with any number of vertices using Python code. We also investigate the notion of a melodic graph, demonstrating how these numbers might impact and motivate musical compositions by employing decagonal numbers as edge labels, exposing intriguing relationships between graph theory and music.