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Gain graphs are graphs where the edges are given some orientation and labeledwith the elements (called gains) from a group so that gains are inverted whenwe reverse the direction of the edges. Generalizing the notion of gain graphs,skew gain graphs have the property that the gain of a reversed edge is theimage of edge gain under an anti-involution. In this paper, we deal with theadjacency matrix of skew gain graphs with involutive automorphism on a field ofcharacteristic zero and their charactersitic polynomials. Spectra of someparticular skew gain graphs are also discussed. Meanwhile it is interesting tonote that weighted graphs are particular cases of skew gain graphs.

On the Characteristic Polynomial of Skew Gain Graphs