<![CDATA[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.]]>
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