Laplacian spectrum characterization of extensions of vertices of wheel graphs and multi-fan graphs

The graph Cn1∇KkCn1∇Kk is the product of a circuit Cn1Cn1 and a clique KkKk. In this paper, we will prove that it is determined by their Laplacian spectrum except when n1=6n1=6. If n1=6n1=6, there are several counterexamples. We also prove that the product of ss vertex–disjoint paths and a clique (Pn1∪Pn2∪⋯∪Pns)∇Kk(Pn1∪Pn2∪⋯∪Pns)∇Kk is also determined by the Laplacian spectrum.