Some Bounds on the Energy of Graphs with Self-Loops regarding λ₁ and λₙ

Let GS 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 GS is defined by Gutman et al. as E(GS)=∑ᵢ₌₁ⁿ| λᵢ -σ/n |, where λ₁,,λₙ are the adjacency eigenvalues of GS and σ is the number of self-loops of GS. In this paper, several upper and lower bounds of E(GS) regarding λ₁ and λₙ are obtained. Especially, the upper bound E(GS) ≤ √n(2m+σ-σ²n) () given by Gutman et al. is improved to the following bound {align*} E(GS)≤ √{n(2m+σ-{σ²}{n})-n/2( |λ₁-σ/n |- |λₙ-σ/n |)²}, {align*} where | λ₁-σ/n| ≥ ≥ | λₙ-σ/n|. Moreover, all graphs are characterized when the equality holds in Gutmans' bound () by using this new bound.