Xue Ding, Tiefeng Jiang, SPECTRAL DISTRIBUTIONS OF ADJACENCY AND LAPLACIAN MATRICES OF RANDOM GRAPHS, The Annals of Applied Probability, Vol. 20, No. 6 (December 2010), pp. 2086-2117 ...

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.