Trace of the Adjacency Matrix of the Star Graph and Complete Bipartite Graph Raised to a Positive Integer Power - Corry Corazon Marzuki; Fitri Aryani; Sri Basriati; Yuslenita Muda (Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang, 2025)
This research aims to derive the general form of the trace matrix of adjacency from star graphs and complete bipartite graphs with size n × n and raised to a positive integer power. To obtain the general form of the trace matrix of adjacency for these graphs, we first derive the general form of the adjacency matrix raised to a positive integer power for each given graph. The general form 14 of matrix exponentiation is proven using mathematical induction. The trace matrix of adjacency for each graph raised to a positive integer power is obtained through a direct proof based on the definition of the trace matrix. Additionally, applications of the trace matrix of adjacency from star graphs and complete bipartite graphs with size n × n and raised to a positive integer power are provided in the form of examples.