Abstract
For natural numbers k<n we study the graphs Tn,k:=Kk∨Kn−k‾. For k=1, Tn,1 is the star Sn−1. For k>1 we refer to Tn,k as a graph of pyramids. We prove that the graphs of pyramids are determined by their spectrum, and that a star Sn is determined by its spectrum iff n is prime. We also show that the graphs Tn,k are completely positive iff k≤2.
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