Laplacian Spectrum and Energy of the Cyclic Order Product Prime Graph of Semi-dihedral Groups

This paper introduces cyclic order product prime graphs to examine the spectral graph properties of semi-dihedral groups, specifically focusing on the Laplacian spectrum and energy. Combining principles from cyclic graphs and order product prime graphs enhances understanding of group algebraic structures. In this context, the cyclic order product prime graph for a finite group is defined as one where two distinct vertices are adjacent if their generated subgroup is cyclic, and the product of their orders is a prime power. Our methodological approach begins with establishing a general presentation for these graphs within semi-dihedral groups. This foundational step is essential for deriving some properties, such as vertex degrees, the number of edges, and Laplacian characteristic polynomials. This information subsequently facilitates the determination of the Laplacian spectrum, characterized by seven eigenvalues of various multiplicities, and the computation of their Laplacian energy. The semi-dihedral group of order 16 is a particular example to illustrate the practicality and generality of our theorems.