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

Let $G$ be a simple graph with vertex set $V=\{v_1,v_2,\dots ,v_n\}$ and ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ the eigenvalues of the adjacency matrix $A$ of $G$. The Estrada index of $G$ is defined as ${\sum }_{k=1}^n{e}^{{\lambda }_k}$. The subgraph centrality of the vertex $v_i$ with respect to $G$ is defined as the $i$th diagonal entry of the matrix $e^{\beta A}$, where $\beta >0$. Let $G^{\sigma }$ be the oriented graph of $G$ with an orientation $\sigma$ and ${\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n$ the eigenvalues of the skew-adjacency matrix of $G^{\sigma }$. The skew Estrada index of $G^{\sigma }$ is defined as ${\sum }_{k=1}^n{e}^{i{\zeta }_k}$. Gao et al. obtained some Coulson-type integral formulas for the Estrada index of $G$ and for the skew Estrada index of $G^{\sigma }$. 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 $G^{\sigma }$, 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 $G^{\sigma }$.