On the seidel matrix of threshold graphs

Abstract

Threshold graph has an important role in graph theory and several applied arreas such as computer science, scheduling theory etc. Here threshold graphs with its binary string representation are considered. Let G be a connected threshold graph with adjacency and Seidel matrix A and S respectively. Then S = J − I − 2A. We study the spectral properties of S. A recurrence formula for characteristic polynomial of S, multiplicity of the eigenvalues ±1 of S and eigenvalue bounds are obtained. Characterisation of threshold graphs with at most five distinct Seidel eigenvalue is shown also. We obtain several bounds on Seidel energy of G. It is shown that our bound is better than Haemers’ bound in practical. Finally, we prove a very uncommon result for threshold graphs: two threshold graphs may be cospectral on Seidel matrix. Here we define a class of such threshold graphs