Toughness and spectral radius in graphs

Let t be a positive integer, and let G be a connected graph of order n with n≥ t+2. A graph G is said to be 1/t-tough if |S|≥1/tc(G-S) for every subset S of $V(G)$ with c(G-S)≥2, where $c(G-S)$ is the number of connected components in $G-S$. The adjacency matrix of G is denoted by $A(G)$. Let λ₁(G)≥λ₂(G)≥≥λₙ(G) be the eigenvalues of $A(G)$. In particular, the eigenvalue λ₁(G) is called the spectral radius of G. In this paper, we prove that G is a 1/t-tough graph unless G=K₁(Kₙ₋ₜ₋₁∪ tK₁) if λ₁(G)≥η(t,n), where η(t,n) is the largest root of x³-(n-t-2)x²-(n-1)x+t(n-t-2)=0.