Laplacian Spectrum of Some Special Graphs

In this work, we study Laplacian spectrum of specific graph families constructed by applying join (Formula: see text) and direct product (×) operators on the complete graph (Formula: see text), complete bipartite graph (Formula: see text), hypercube graph (Formula: see text) and cycle graph (Formula: see text) for some integer Formula: see text. In particular, we derive complete Laplacian spectrum of the following graphs explicitly and determine for which Formula: see text they can be calculated: (i) Formula: see text, (ii) Formula: see text, (iii) Formula: see text, (iv) Formula: see text. We note that the graphs Formula: see text are the regular graphs whose Laplacian spectrum are known. By applying graph operations on these graphs, we extend the existing knowledge of Laplacian spectra. Our results suggest that further exploration of the Laplacian spectra of other graph families, using similar graph operations, could yield valuable insights.