Eigenvalues and energy in threshold graphs

Assuming a uniform random model of selecting creation sequences, we show that almost every connected threshold graph has more negative than positive eigenvalues. We show that no threshold graphs have eigenvalues in (−1,0)(−1,0). Sequences of equienergetic graphs are given including the striking result that for all n≥3n≥3, there exist n−1n−1 threshold graphs of order n2n2, pairwise noncospectral, each equienergetic to Kn2Kn2. For all n>8n>8, we find an n-vertex hyperenergetic threshold graph.