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1.
The Waring problem for matrix algebras, II
Matej Brešar, Peter Šemrl, 2023, original scientific article

Abstract: Let $f$ be a noncommutative polynomial of degree $m\ge 1$ over an algebraically closed field $F$ of characteristic $0$. If $n\ge m-1$ and $\alpha_1,\alpha_2,\alpha_3$ are nonzero elements from $F$ such that $\alpha_1+\alpha_2+\alpha_3=0$, then every trace zero $n\times n$ matrix over $F$ can be written as $\alpha_1 A_1+\alpha_2A_2+\alpha_3A_3$ for some $A_i$ in the image of $f$ in $M_n(F)$.
Keywords: Waring problem, noncommutatative polynomials, matrix algebras
Published in DiRROS: 10.04.2024; Views: 93; Downloads: 41
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2.
Zhang-Zhang polynomials of phenylenes and benzenoid graphs
Niko Tratnik, 2024, original scientific article

Abstract: The aim of this paper is to study some variations of the Zhang-Zhang polynomial for phenylenes, which can be obtained as special cases of the multivariable Zhang-Zhang polynomial. Firstly, we prove the equality between the first Zhang-Zhang polynomial of a phenylene and the generalized Zhang-Zhang polynomial of some benzenoid graph, which enables us to prove also the equality between the first Zhang-Zhang polynomial and the generalized cube polynomial of the resonance graph. Next, some results on the roots of the second Zhang-Zhang polynomial of phenylenes are provided and another expression for this polynomial is established. Finally, we give structural interpretation for (partial) derivatives of different Zhang-Zhang polynomials.
Keywords: graph theory, resonance graphs, polynomials
Published in DiRROS: 18.03.2024; Views: 105; Downloads: 36
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