Let a finite group G of odd order n act regularly on a connected (multi-)graph Γ. That is, no group element other than the identity fixes any vertex. Then the “quotient graph” Δ under the action is the induced graph of orbits. We give a result about the connectivity of Γ and Δ in terms of their numbers of labeled spanning trees. In words, the spanning tree count of the graph is equal to n, the order of the given regular automorphism group, times the spanning tree count of the graph of orbits, times a perfect square integer. There is a dual result on the Laplacian spectrum saying that the multiset of Laplacian eigenvalues for the main graph is the disjoint union of the multiset for the quotient graph together with a multiset all of whose elements have even multiplicity. Specializing to the case of one orbit, we observe that a Cayley graph of odd order has spanning tree count equal to n times a square, and that that the Laplacian spectrum consists of the value 0 together with other doubled eigenvalues. These results are based on a study of matrices (and determinants) that consist of blocks of group-matrices. The generic determinant for such a matrix with the additional property of symmetry will have a dominanting square factor in its (multinomial) factorization. To show this, we make use of the Feit-Thompson theorem which provides a normal tower for an odd-order group, and perform a similarity conjugation with a fixed integer, unimodal matrix. Additional related results are given for certain group-matrices “twisted” by a group of automorphisms, generalizing the “g-circulants” of P.J. Davis.
Resultants and group matrices
