On the inertia of weighted (k−1)-cyclic 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, respectively, the number of the positive, negative and zero eigenvalues of the adjacency matrix A(Gw) of Gw including their multiplicities. A simple n-vertex connected graph is called a (k − 1)-cyclic graph if its number of edges equals n + k − 2. Let θ(r1, r2, . . . , rk)w be an n-vertex simple weighted graph obtained from k weighted paths (Pr1)w, (Pr2)w, . . . , (Prk)w by identifying their initial vertices and terminal vertices, respectively. Set Θk := {θ(r1, r2, . . . , rk)w : r1 + r2 + · · · + rk = n + 2k − 2}. The inertia of the weighted graph θ(r1, r2, . . . , rk)w is studied. Also, the weighted (k − 1)-cyclic graphs that contain θ(r1, r2, . . . , rk)w as an induced subgraph are studied. We characterize those graphs among Θk that have extreme inertia. The results generalize the corresponding results obtained by Tan and Liu in 2013 and Yu et al., 2014.