The spectral radii and extremal graphs of two types of minimal graphs

A connected nontrivial graph G is matching covered if every edge of G is contained in some perfect matching of G . A matching covered graph G is minimal if 𝐺 − 𝑒 is not matching covered for each edge e of G . A graph is said to be factor-critical if 𝐺 − 𝑣 has a perfect matching for every 𝑣 ∈ 𝑉 ⁡ ( 𝐺 ) . A factor-critical graph G is said to be minimal factor-critical if 𝐺 − 𝑒 is not factor-critical for each edge 𝑒 ∈ 𝐸 ⁡ ( 𝐺 ) . In this paper, by employing ear decomposition and edge-exchange techniques, the greatest spectral radii of minimal matching covered bipartite graphs and minimal factor-critical graphs are determined, and the corresponding extremal graphs are characterized.