The extremal problems on the inertia of weighted bicyclic graphs

LetGw be a weighted graph. The number of the positive, negative and zero eigenvalues in the spectrum of Gw are called positive inertia index, negative inertia index and nullity of Gw, and denoted by i+(Gw), i−(Gw), i0(Gw), respectively. In this paper, sharp lower bound on the positive (resp. negative) inertia index of weighted bicyclic graphs of order n with pendant vertices is obtained. Moreover, all the weighted bicyclic graphs of order n with at most two positive, two negative and at least n− 4 zero eigenvalues are identified, respectively.