Laplacian spectral metrics of some graph operations

The Laplacian spectral metrics (including the Laplacian energy, Kirchhoff index, Laplacian-energy-like invariant and Laplacian entropy.) serve as important mathematical tools and are widely applied in various research fields including theoretical chemistry, bioinformatics, and network science. In this paper, we study the Laplacian spectral metrics for [Formula: see text] thorny graphs, [Formula: see text] splitting graphs, and [Formula: see text] shadow graphs, and give their expressions, respectively.