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

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Abstract | References | Similar Articles | Additional Information

Abstract: Let ${A_1}, \ldots ,{A_k}$ be $n \times n$ matrices over a commutative ring $R$ with identity. Graph theoretic methods are established to compute the standard polynomial $[{A_1}, \ldots ,{A_k}]$. It is proved that if $k < 2n - 2$, and if the characteristic of $R$ either is zero or does not divide $4I(1/2n) - 2$, where $I$ denotes the greatest integer function, then there exist $n \times n$ skew-symmetric matrices ${A_1}, \ldots ,{A_k}$ such that $[{A_1}, \ldots ,{A_k}] \ne 0$.

- A. S. Amitsur and J. Levitzki,
*Minimal identities for algebras*, Proc. Amer. Math. Soc.**1**(1950), 449–463. MR**36751**, DOI https://doi.org/10.1090/S0002-9939-1950-0036751-9
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*Research Problems: Identities on Matrices*, Amer. Math. Monthly**79**(1972), no. 2, 157–158. MR**1536623**, DOI https://doi.org/10.2307/2316538 - Richard G. Swan,
*An application of graph theory to algebra*, Proc. Amer. Math. Soc.**14**(1963), 367–373. MR**149468**, DOI https://doi.org/10.1090/S0002-9939-1963-0149468-6 - Richard G. Swan,
*Correction to “An application of graph theory to algebra”*, Proc. Amer. Math. Soc.**21**(1969), 379–380. MR**255439**, DOI https://doi.org/10.1090/S0002-9939-1969-0255439-7

*A graph theoretic generalization of a theorem by Kostant*(to appear). ---,

*Matrices with zero diagonal*, Notices Amer. Math. Soc.

**20**(1973), A-7 and A-548. Abstract 73T-A25.

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

Article copyright:
© Copyright 1975
American Mathematical Society