Each (n,m)-graph having the i-th minimal Laplacian coefficient is a threshold graph

Abstract

Let G be a simple graph with n vertices and m edges (i.e. an (n,m)-graph) and L(G) be the Laplacian matrix of G. The Laplacian characteristic polynomial of G is defined as P(G;λ)=det⁡(λI−L(G))=∑i=0n(−1)ici(G)λn−i, where ci(G) is referred as the i-th Laplacian coefficient of G. Denote Gn,m by the set of all connected (n,m)-graphs. A connected graph H∈Gn,m is called ci-minimal if ci(H)≤ci(G) holds for each G∈Gn,m and is called uniformly minimal if H is ci-minimal for i=0,1,…,n. In this paper, we prove that each ci-minimal graph in Gn,m is a threshold graph for 2≤i≤n−2. Moreover, we prove that there does not exist uniformly minimal graphs in Gn,n+3, n≥6.