Coulson-Type integral formulas for the estrada index of graphs and the skew estrada index of oriented graphs

The energy of graphs and skew energy of oriented graphs have been studied extensively in recent years. The Estrada index, an invariant of graphs, which is similar to the energy, has also received much attention nowadays. Motivated by these three graph invariants, in this paper, we introduce a new one, the skew Estrada index of oriented graphs. For an oriented graph Gσ with n vertices, its skew Estrada index is defined as EE_s(Gσ) = ∑_nk=1e_iλk, where λ₁; λ₂;.;λ_n are the eigenvalues of Gσ. A well-known result on the energy of graphs is the Coulson integral formula which was given by Coulson in 1940. Our main results in this paper are some integral formulas for the Estrada index of graphs and the skew Estrada index of oriented graphs, which are counterparts of the Coulson integral formula for the energy of graphs.