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Graph energies and spectral radii have application in molecular computing and computer sciences. In this paper, we concentrate on new findings about the Symmetric Division Deg, (SDD) energies, and (SDD) spectral radii of p splitting and p shadow graphs built on the foundation of any regular graph. In fact it becomes natural to ask how the SDD energies and spectral radii of newly formed graphs depend on the SDD energies and spectral radii of the given regular graph. [ABSTRACT FROM AUTHOR]

An eigenvalue λ of a graph G of order n is a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector j n. In 1978, Cvetković proved that G has exactly one main eigenvalue if and only if G is regular, and posed the following long-standing problem: characterize the graphs with exactly k (2 ≤ k ≤ n) main eigenvalues. Graphs of order n with n , n − 1 main eigenvalues are called controllable, almost controllable, respectively. Cographs, threshold graphs are frequently studied in structural graph theory and computer science. In this paper, all almost controllable cographs, all almost controllable threshold graphs and all almost controllable graphs with second largest eigenvalue less than or equal to 5 − 1 2 are characterized. Furthermore, we give some results about cographs with exactly n − 2 main eigenvalues, and propose some additional problems for further study. [ABSTRACT FROM AUTHOR]