Toughness and A{α}-spectral radius in graphs

Let α∈[0,1), and let G be a connected graph of order n with n≥ f(α), where f(α)=6 for α∈[0,2/3] and f(α)=4/1-α for α∈(2/3,1). A graph G is said to be t-tough if |S|≥ tc(G-S) for each subset S of $V(G)$ with c(G-S)≥2, where $c(G-S)$ is the number of connected components in $G-S$. The Aα-spectral radius of G is denoted by ρα(G). In this paper, it is verified that G is a 1-tough graph unless G=K₁(Kₙ₋₂∪ K₁) if ρα(G)≥ρα(K₁(Kₙ₋₂∪ K₁)), where ρα(K₁(Kₙ₋₂∪ K₁)) equals the largest root of x³-((α+1)n+α-3)x²+(α n²+(α²-α-1)n-2α+1)x-α²n²+(3α²-α+1)n-4α²+5α-3=0. Further, we present an Aα-spectral radius condition for a graph to be a t-tough graph.