Relations between degrees, conjugate degrees and graph energies

Abstract

Let G be a simple graph of order n with maximum degree Δ and minimum degree δ. Let (d)=(d1,d2,…,dn) and (d⁎)=(d1⁎,d2⁎,…,dn⁎) be the sequences of degrees and conjugate degrees of G. We define π=∑i=1ndi and π⁎=∑i=1ndi⁎, and prove that π⁎≤LEL≤IE≤π where LEL and IE are, respectively, the Laplacian-energy-like invariant and the incidence energy of G. Moreover, we prove that π−π⁎>(δ/2)(n−Δ) for a certain class of graphs. Finally, we compare the energy of G and π, and present an upper bound for the Laplacian energy in terms of degree sequence.