Graphs with the minimal Laplacian coefficients

Let (Formula presented.) be the Laplacian matrix of a connected graph G of order n. The Laplacian polynomial of G is defined as (Formula presented.), where (Formula presented.) is the ith Laplacian coefficient of G. A graph (Formula presented.), which is the set of all connected graphs of order n with m edges, is called (Formula presented.) -minimal if (Formula presented.), (Formula presented.), (Formula presented.). Furthermore, H is called uniformly minimal in (Formula presented.) if H is (Formula presented.) -minimal for (Formula presented.). In this paper, we prove that when (Formula presented.), the threshold graph (Formula presented.) is the unique uniformly minimal graph in (Formula presented.) except for the cases (Formula presented.) and (Formula presented.). Moreover, (Formula presented.) is also the unique uniformly minimal graph for (Formula presented.).