Modular Multiplicative Divisor Labeling Techniques for Efficient Minimum Spanning Tree Computation in Jellyfish Graph Using Kruskal Algorithm

This paper investigates minimum spanning tree of Jellyfish graph admits Modular Multiplicative Divisor (MMD) labeling. The Jellyfish graph is constructed by expanding a cycle <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$C_{4}$</tex> with attached pendant paths, forming a structure that resembles a jellyfish. MMD labeling assigns unique natural numbers to each vertex, with edge labels derived from the product of vertex labels modulo <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex>. The study explores MMD labeling within minimum spanning tree of Jellyfish graphs with an even number of tentacles. MST is defined as spanning trees with the minimum possible total edge weight in a weighted graph. The Minimum Spanning Tree contains all vertices while minimizing edge weight sum without forming cycles, ensuring efficient connectivity. The findings contribute to an enhanced understanding of MMD labeling applications in complex graph configurations.