Generating Functions and the Minimum Spectral Radius in Strongly Connected Digraphs with $m+2$ Edges

We study the minimum adjacency spectral radius in the class $\mathcal{SC}_{m+2}(m)$ of strongly connected digraphs with $m$ vertices and $m+2$ edges. Using generating functions for directed paths, we associate with the relevant digraphs topological polynomials whose smallest positive roots determine