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 G with n vertices, its skew Estrada index is dened as

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