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SPEKTRUM GRAF PANGKAT ATAS GRUP DIHEDRAL
The power graph is a graph that represents the power relation between elements within a group, where two distinct vertices are connected by an edge if one element is a power of the other. This study focuses on determining the spectrum of the power graph over the dihedral group D2n restricted to the orders n = pk and n = pq, where p and q are distinct prime numbers and k is a natural number. In the computation, the process begins by constructing the power graph of the dihedral group D2n based on the power relations among its elements. Once the graph structure and the degree of each vertex are identified, the power graph is represented in the form of an adjacency matrix, Laplacian matrix, signless Laplacian matrix, as well as the normalized forms of these three representation matrices. From each representation matrix, the characteristic polynomial equation is constructed to find the eigenvalues along with their multiplicities, collectively referred to as the spectrum of the graph. During the matrix simplification and the computation of eigenvalues, identifying the element orders and generating subgroups becomes highly crucial. In the case of n = pk, the subgroups form an ordered chain, allowing for the systematic calculation of vertex degrees and the determination of eigenvalues. Conversely, in the case of n = pq, the subgroup structure exhibits branching, leading to high complexity in calculating the vertex degrees. Consequently, the determination of the spectrum for the n = pq case is specifically limited to the adjacency matrix. The results of this research formulate the characteristic polynomials along with the eigenvalues of the corresponding graph representation matrices for each case of the power graph on the dihedral group.
FAKULTAS SAINS DAN TEKNOLOGI, UIN SUNAN KALIJAGA YOGYAKARTA
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