Applications of Graph Theory to Matrix Theory

Abstract

Let <!-- MATH ${A_1}, \ldots ,{A_k}$ --> be <!-- MATH $n \times n$ --> matrices over a commutative ring with identity. Graph theoretic methods are established to compute the standard polynomial <!-- MATH $[{A_1}, \ldots ,{A_k}]$ --> . It is proved that if <!-- MATH $k < 2n - 2$ --> <img width="100" height="39" align="MIDDLE" border="0" src="images/img5.gif" alt="$ k < 2n - 2$">, and if the characteristic of either is zero or does not divide <!-- MATH $4I(1/2n) - 2$ --> , where denotes the greatest integer function, then there exist <!-- MATH $n \times n$ --> skew-symmetric matrices <!-- MATH ${A_1}, \ldots ,{A_k}$ --> such that <!-- MATH $[{A_1}, \ldots ,{A_k}] \ne 0$ --> .