On Graphs with c2-c3 Successive Minimal Laplacian Coefficients

Abstract

Let G be a graph of order n and L(G) be its Laplacian matrix. The Laplacian polynomial of G is defined as P(G;λ)=det(λI−L(G))=∑i=0n(−1)ici(G)λn−i, where ci(G) is called the i-th Laplacian coefficient of G. Denoted by Gn,m the set of all (n,m)-graphs, in which each of them contains n vertices and m edges. The graph G is called uniformly minimal if, for each i(i=0,1,…,n), H is ci(G)-minimal in Gn,m. The Laplacian matrix and eigenvalues of graphs have numerous applications in various interdisciplinary fields, such as chemistry and physics. Specifically, these matrices and eigenvalues are widely utilized to calculate the energy of molecular energy and analyze the physical properties of materials. The Laplacian-like energy shares a number of properties with the usual graph energy. In this paper, we investigate the existence of uniformly minimal graphs in Gn,m because such graphs have minimal Laplacian-like energy. We determine that the c2(G)-c3(G) successive minimal graph is exactly one of the four classes of threshold graphs.