Coulson-type integral formulas for the general (skew) Estrada index of a vertex

Let G be a simple graph with vertex set V={v₁,v₂,…,v_n} and λ₁,λ₂,…,λ_n the eigenvalues of the adjacency matrix A of G. The Estrada index of G is defined as ∑_k=1ⁿe^λ_k. The subgraph centrality of the vertex v_i with respect to G is defined as the ith diagonal entry of the matrix e^βA, where β>0. Let G^σ be the oriented graph of G with an orientation σ and ζ₁,ζ₂,…,ζ_n the eigenvalues of the skew-adjacency matrix of G^σ. The skew Estrada index of G^σ is defined as ∑_k=1ⁿe^iζ_k. Gao et al. obtained some Coulson-type integral formulas for the Estrada index of G and for the skew Estrada index of G^σ. In this paper, we will introduce the concept of the general Estrada index of v_i with respect to G as a generalization of subgraph centrality and the concept of the general skew Estrada index of v_i with respect to G^σ, and give some Coulson-type integral formulas for the general vertex Estrada index with respect to G and for the general vertex skew Estrada index with respect to G^σ.