A sharp lower bound for the spectral radius in K₄-saturated graphs

For given graphs G and H, the graph G is H-saturated if G does not contain H as a subgraph but for any e∈E(G‾), G+e contains H. In this note, we prove that if G is an n-vertex K_r+1-saturated graph such that for each vertex v∈V(G), ∑w∈N(v)d_G(w)≥(r−2)d(v)+(r−1)(n−r+1), then ρ(G)≥ρ(S_n,r), where S_n,r is the graph obtained from a copy of K_r−1 with vertex set S by adding n−r+1 vertices, each of which has neighborhood S. This provides a sharp lower bound for the spectral radius in an n-vertex K_r+1-saturated graph for r=2,3, verifying a special case of a conjecture by Kim, Kim, Kostochka and O.