Some bounds on the spectral radius of connected threshold graphs

The spectral radius of a graph is the spectral radius of its adjacency matrix. A threshold graph is a simple graph whose vertices can be ordered as $v_1, v_2, \ldots, v_n$, so that for each $2 \le i \le n$, vertex $v_i$ is either adjacent or nonadjacent simultaneously to all of $v_1, v_2, \ldots, v_{i-1}$. Brualdi and Hoffman initially posed and then partially solved the extremal problem of finding the simple graphs with a given number of edges that have the maximum spectral radius. This problem was subsequently completely resolved by Rowlinson. Here, we deal with the similar problem of maximizing the spectral radius over the set of connected simple graphs with a given number of vertices and edges. As shown by Brualdi and Solheid, each such extremal graph is necessarily a threshold graph. We investigate the spectral radii of threshold graphs by relying on computations involving lazy walks. Furthermore, we obtain three lower bounds and one upper bound on the spectral radius of a given connected threshold graph.