| Summary: | This paper introduces cyclic order product prime graph 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. Let G be a finite group. In this context, the cyclic order product prime graph of G is defined as a simple undirected graph with vertex set G where two distinct vertices, x and y, are adjacent if and only if 〈x, y〉 is a proper cyclic subgroup of G and |x||y| = pα, α ∈ ℕ for some prime p. 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 serves as a particular example to illustrate the practicality and generality of our theorems. ©Copyright Mohamed.
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