The inertia of weighted unicyclic graphs

Let Gw be a weighted graph. The inertia of Gw is the triple In(Gw)=(i+(Gw),i−(Gw),i0(Gw)), where i+(Gw), i−(Gw), i0(Gw) are the numbers of the positive, negative and zero eigenvalues of the adjacency matrix A(Gw) of Gw including their multiplicities, respectively. i+(Gw), i−(Gw) are called the positive, negative indices of inertia of Gw, respectively. In this paper we present a lower bound for the positive, negative indices of weighted unicyclic graphs of order n with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound. Moreover, we characterize the weighted unicyclic graphs of order n with two positive, two negative and at least n−6 zero eigenvalues, respectively.