Applications of graph theory to matrix theory

Author:
Frank W. Owens

Journal:
Proc. Amer. Math. Soc. **51** (1975), 242-249

MSC:
Primary 15A15

DOI:
https://doi.org/10.1090/S0002-9939-1975-0376708-9

MathSciNet review:
0376708

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be matrices over a commutative ring with identity. Graph theoretic methods are established to compute the standard polynomial . It is proved that if , and if the characteristic of either is zero or does not divide , where denotes the greatest integer function, then there exist skew-symmetric matrices such that .

**[1]**S. A. Amitsur and J. Levitzki,*Minimal identities for algebras*, Proc. Amer. Math. Soc.**1**(1950), 449-463. MR**12**, 155. MR**0036751 (12:155d)****[2]**Joan P. Hutchinson, Doctoral Thesis, University of Pennsylvania, 1973.**[3]**Bertram Kostant,*A theorem of Frobenius, a theorem of Amitsur-Levitzki and cohomology theory*, J. Math. Mech.**7**(1958), 237-264. MR**19**, 1153. MR**0092755 (19:1153e)****[4]**Frank W. Owens,*A graph theoretic generalization of a theorem by Kostant*(to appear).**[5]**-,*Matrices with zero diagonal*, Notices Amer. Math. Soc.**20**(1973), A-7 and A-548. Abstract 73T-A25.**[6]**Louis H. Rowen,*Standard polynomials in matrix algebras*, Trans. Amer. Math. Soc.**190**(1974), 253-284. MR**0349715 (50:2208)****[7]**Kirby C. Smith and Hillel J. Kumin,*Identities on matrices*, Amer. Math. Monthly**79**(1972), 157-158. MR**1536623****[8]**Richard G. Swan,*An application of graph theory to algebra*, Proc. Amer. Math. Soc.**14**(1963), 367-373. MR**26**#6956. MR**0149468 (26:6956)****[9]**-,*Correction to ``An application of graph theory to algebra"*, Proc. Amer. Math. Soc.**21**(1969), 379-380. MR**41**#101. MR**0255439 (41:101)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
15A15

Retrieve articles in all journals with MSC: 15A15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1975-0376708-9

Keywords:
Standard polynomial,
digraph,
Euler path,
skew-symmetric

Article copyright:
© Copyright 1975
American Mathematical Society