Graphs with the minimal Laplacian coefficients

Abstract

Let L ( G ) be the Laplacian matrix of a connected graph G of order n. The Laplacian polynomial of G is defined as P ( G ; λ ) = det ( λI − L ( G ) ) = ∑ i = 0 n ( − 1 ) i c i ( G ) λ n − i , where c i ( G ) is the ith Laplacian coefficient of G. A graph H ∈ G n , m , which is the set of all connected graphs of order n with m edges, is called c i -minimal if c i ( H ) ≤ c i ( G ) , 0 ≤ i ≤ n , ∀ G ∈ G n , m . Furthermore, H is called uniformly minimal in G n , m if H is c i -minimal for i = 0 , 1 , … , n . In this paper, we prove that when m ≥ ( n 2 ) − ( 2 n − 5 ) , the threshold graph L n , m is the unique uniformly minimal graph in G n , m except for the cases ( n , m ) = ( 6 , 9 ) and ( 8 , 18 ) . Moreover, L n , m is also the unique uniformly minimal graph for 3 n 2 − 8 n + 5 8 < m ≤ ( n 2 ) .