Relations between degrees, conjugate degrees and graph energies.

Let G be a simple graph of order n with maximum degree Δ and minimum degree δ . Let ( d ) = ( d 1 , d 2 , … , d n ) and ( d ⁎ ) = ( d 1 ⁎ , d 2 ⁎ , … , d n ⁎ ) be the sequences of degrees and conjugate degrees of G . We define π = ∑ i = 1 n d i and π ⁎ = ∑ i = 1 n d i ⁎ , and prove that π ⁎ ≤ L E L ≤ I E ≤ 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. [ABSTRACT FROM AUTHOR]