The maximum -spectral radius of -connected graphs with bounded matching number.

For a graph G, let A(G) be its adjacency matrix and let D(G) be the diagonal matrix of its vertex degrees. For α∈[0,1], Nikiforov [18] introduced the Aα-matrix of G as follows:Aα(G)=αD(G)+(1−α)A(G). Let n, t, k be positive integers such that t≥1, k≥2, n≥k+2, and n≡k (mod 2). In this paper, for α∈[0,12], we prove sharp upper bounds for spectral radius of Aα(G) in an n-vertex t-connected graph with the matching number at most n−k2. This result extends the one of O (2021) [21] and the one of Zhang (2022) [25].