Optimized Modular Multiplicative Divisor Labeling for Efficient Minimum Spanning Tree Computation in Jellyfish Graphs Using Prims Algorithm

The Minimum Spanning Tree (MST) of the Jellyfish graph, which admits Modular Multiplicative Divisor (MMD) labeling, is analyzed using Prim's Algorithm. The Jellyfish graph extends a cycle <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{C}_{\mathbf{4}}$</tex> with pendant paths, resembling a jellyfish. Modular Multiplicative Divisor labeling assigns unique natural numbers to vertices, with edge labels derived modulo <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{n}$</tex>. Prim's algorithm efficiently computes the MST by selecting the lowest weight edges while ensuring connectivity, acyclicity, and minimal total weight. The study explores <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{J}(\boldsymbol{\theta},\boldsymbol{\omega})$</tex> graphs with even tentacles, highlighting the role of Modular Multiplicative Divisor labeling in optimizing spanning tree structures for network applications.