Characteristic polynomials of some graph coverings
Finite simple undirected graphs \(G = G(V,E)\) and their adjacency matrices \(A(G)\) are considered. By the characteristic polynomial \(\Phi (G; \lambda)\) of \(G\) we understand the determinant of \(\lambda I - A(G)\). Let \(D(G)\) be the set of the \(2 |E |\) directed edges (between elements of \(V)\) such that we take \((u,v)\) and \((v,u)\) for any element \([u,v]\) of \(E\). The authors introduce a graph \(G^\alpha\), called the derived graph of \((G, \alpha)\) [or a derived graph covering of \(G\) with voltages in \(\Gamma]\), where \(\alpha\) is a mapping of \(D(G)\) to a finite group \(\Gamma\) and \(\alpha (v,u) = (\alpha (u,v))^{-1}\) is postulated for any pair of adjacent vertices of \(G\). It is shown that \(\Phi (G^\alpha; \lambda)\) is divisible by \(\Phi (G; \lambda)\), the quotient polynomial is determined in terms of the irreducible representation of \(\Gamma\) and the Kronecker product of matrices.
scientific article; zbMATH DE number 790769