On the Various Energy Forms of the Join of Complete Graphs

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The concept of graph energy, first introduced in 1978, has been a focal point of extensive research within the field of graph theory, leading to the publication of numerous articles. Graph energy, originally associated with the eigenvalues of the adjacency matrix of a graph, has since been extended to various other matrices. These include the maximum degree matrix, Randić matrix, sum-connectivity matrix, and the first and second Zagreb matrices, among others. In this paper, we focus on calculating the energy of several such matrices for the join graph of complete graphs, denoted as J m ( K n ) . Specifically, we compute the energies for the maximum degree matrix, Randić matrix, sum-connectivity matrix, first Zagreb matrix, second Zagreb matrix, reverse first Zagreb matrix, and reverse second Zagreb matrix for J m ( K n ) . Our results provide new insights into the structural properties of the join graph and contribute to the broader understanding of the mathematical characteristics of graph energy for different matrix representations. This work extends the scope of graph energy research by considering these alternative matrix forms, offering a deeper exploration into the algebraic and spectral properties of graph energy in the context of join graphs.