On energy and Laplacian energy of chain graphs

Let G be a simple graph of order n. The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. The Laplacian energy of the graph G is defined as LE=LE(G)=∑i=1nμ_i−d¯,where μ₁,μ₂,…,μ_n−1,μ_n=0 are the Laplacian eigenvalues, and d¯ is the average degree of graph G. In this paper we present some lower and upper bounds on E(G) of chain graph G. From this we prove that the star gives the minimal energy of connected chain graphs of order n. We present a lower bound on LE(G) of chain graphs G in terms of order n, and characterize the extremal graphs. Moreover, we obtain the maximal Laplacian energy among all connected chain graphs of order n with n edges. Finally, we propose an open problem on Laplacian energy of chain graphs.