Some New Results on Energy of Graphs with Self Loops

Abstract

The graph Gσ is obtained from graph G by attaching self loops on σ vertices. The energy E(Gσ) of the graph Gσ with order n and eigenvalues λ1, λ2, . . . , λn is defined as E(Gσ) = Xn i=1 λi −σ/n.It has been proved that if σ = 0 or n then E(G) = E(Gσ).The obvious question arise: Are there any graph such that E(G) =E(Gσ) and 0< σ < n? We have found an affirmative answer of thisquestion and contributed a graph family which satisfies this property.