Alan J. Hoffman, The Change in the Least Eigenvalue of the Adjacency Matrix of a Graph Under Imbedding, SIAM Journal on Applied Mathematics, Vol. 17, No. 4 (Jul., 1969), pp. 664-671 ...

Let G be a graph of order n and let S be any subset of the vertex set V (G). Denote by GS the graph obtained from the graph G by adding a new vertex x which is adjacent exactly to the vertices from S.