Other Situations Involving Group Matrices

2019 ◽  
pp. 313-340
Author(s):  
Kenneth W. Johnson
Keyword(s):  
Group Matrices

scholarly journals Resultants and group matrices

1980 ◽  
Vol 33 ◽  
pp. 111-122 ◽  
Author(s):  
Kai Wang
Keyword(s):  
Group Matrices

CONNECTIVITY AND SPECTRUM IN A GRAPH WITH A REGULAR AUTOMORPHISM GROUP OF ODD ORDER

1994 ◽  
Vol 04 (04) ◽  
pp. 529-560 ◽  
Author(s):  
JON A. SJOGREN
Keyword(s):  
Automorphism Group ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Laplacian Spectrum ◽  
Regular Automorphism ◽  
Fixed Integer ◽  
Quotient Graph ◽  
Odd Order ◽  
Induced Graph ◽  
Group Matrices

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.

scholarly journals Unimodular group matrices with rational integers as elements

1964 ◽  
Vol 14 (2) ◽  
pp. 719-726 ◽  
Author(s):  
Robert Thompson
Keyword(s):  
Unimodular Group ◽  
Group Matrices

Minimal rank of abelian group matrices

1996 ◽  
Author(s):  
Yip-cheung Chan
Keyword(s):  
Abelian Group ◽  
Minimal Rank ◽  
Group Matrices

scholarly journals Abelian group matrices over the p-adic and rational integers

1975 ◽  
Vol 7 (1) ◽  
pp. 77-90
Author(s):  
Dennis Garbanati
Keyword(s):  
Abelian Group ◽  
Group Matrices

scholarly journals Modular theory of group-matrices

1907 ◽  
Vol 8 (3) ◽  
pp. 389-389 ◽  
Author(s):  
Leonard Eugene Dickson
Keyword(s):  
Modular Theory ◽  
Group Matrices

A theorem concerning group matrices

1981 ◽  
Vol 9 (8) ◽  
pp. 883-891
Author(s):  
E. Rips
Keyword(s):  
Group Matrices

scholarly journals Group matrices for the quaternion and generalized dihedral groups

1986 ◽  
Vol 12 (5-6) ◽  
pp. 1297-1301
Author(s):  
K. Wang ◽  
P.J. Davis
Keyword(s):  
Dihedral Groups ◽  
Group Matrices
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