The inertia of weighted unicyclic graphs

Let $G_w$ be a weighted graph. The $\mathit{\text{inertia}}$ of $G_w$ is the triple $In(G_w)=(i_{+}(G_w),i_{-}(G_w),$ $i_0(G_w))$, where $i_{+}(G_w),i_{-}(G_w),i_0(G_w)$ are the number of the positive, negative and zero eigenvalues of the adjacency matrix $A(G_w)$ of $G_w$ including their multiplicities, respectively. $i_{+}(G_w)$, $i_{-}(G_w)$ is called the $\mathit{\text{positive,}}\mathit{\text{negative}}\mathit{\text{index}}\mathit{\text{of}}\mathit{\text{inertia}}$ of $G_w$, respectively. In this paper we present a lower bound for the positive, negative index 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.