Some Bounds on the Energy of Graphs with Self-Loops regarding λ_1 and λ_n
arxiv(2024)
Abstract
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.
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