Some Bounds on the Energy of Graphs with Self-Loops regarding λ_1 and λ_n

Let G_S 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 G_S is defined by Gutman et al. as E(G_S)=∑_i=1^n| λ_i -σ/n|, where λ_1,…,λ_n are the adjacency eigenvalues of G_S and σ is the number of self-loops of G_S. In this paper, several upper and lower bounds of E(G_S) regarding λ_1 and λ_n are obtained. Especially, the upper bound E(G_S) ≤√(n(2m+σ-σ^2/n)) (∗) given by Gutman et al. is improved to the following bound E(G_S)≤√(n(2m+σ-σ^2/n)-n/2( |λ_1-σ/n |- |λ_n-σ/n |)^2), where | λ_1-σ/n| ≥…≥| λ_n-σ/n|. Moreover, all graphs are characterized when the equality holds in Gutmans' bound (∗) by using this new bound.