Some Bounds on the Energy of Graphs with Self-Loops regarding ₁ and ₍

Let Gₒ be a graph with n vertices obtained from a simple graph G by attaching one self-loop at each vertex in S V (G). The energy of Gₒ is defined by Gutman et al. as E (Gₒ) =₈=₁^n| ₈ -n |, where ₁, , ₍ are the adjacency eigenvalues of Gₒ and is the number of self-loops of Gₒ. In this paper, several upper and lower bounds of E (Gₒ) regarding ₁ and ₍ are obtained. Especially, the upper bound E (Gₒ) n (2m+-^{2n) } () given by Gutman et al. is improved to the following bound align* E (Gₒ) n (2m+-^{2n) -n2 (|₁-n |- |₍-n |) ^2}, align* where | ₁-n| | ₍-n|. Moreover, all graphs are characterized when the equality holds in Gutmans' bound () by using this new bound.