The \(\mathscr{G}\)-average of adjacency and Laplacian polynomials of a graph

Let \(\mathscr{G}\) be a finite multiplicative group of quaternion unit \(U(\mathbb{H})\). The \(\mathscr{G}\)-average of adjacency (resp., Laplacian) polynomial of a graph \(G\) is defined as the arithmetic mean of the characteristic polynomials of adjacency (resp., Laplacian) matrice of all \(\mathscr{G}\)-gain graphs on \(G\). In this paper, we prove that the \(\mathscr{G}\)-average adjacency (resp., Laplacian) polynomial of a graph for any non-trivial finite subgroup \(\mathscr{G}\) of \(U(\mathbb{H})\) coincides with its matching (resp., weighted TU-subgraph) polynomial, which generalizes previous findings for signed graphs.