| 1. |
- Akbari, Saieed, et al.
(författare)
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Chromatic number and clique number of subgraphs of regular graph of matrix algebras
- 2012
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Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 436:7, s. 2419-2424
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Tidskriftsartikel (refereegranskat)abstract
- Let R be a ring and X subset of R be a non-empty set. The regular graph of X, Gamma(X), is defined to be the graph with regular elements of X (non-zero divisors of X) as the set of vertices and two vertices are adjacent if their sum is a zero divisor. There is an interesting question posed in BCC22. For a field F, is the chromatic number of Gamma(GL(n)(F)) finite? In this paper, we show that if G is a soluble sub-group of GL(n)(F), then x (Gamma(G)) < infinity. Also, we show that for every field F, chi (Gamma(M-n(F))) = chi (Gamma(M-n(F(x)))), where x is an indeterminate. Finally, for every algebraically closed field F, we determine the maximum value of the clique number of Gamma(< A >), where < A > denotes the subgroup generated by A is an element of GL(n)(F). (C) 2011 Elsevier Inc. All rights reserved.
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| 2. |
- Akbari, Saieed, et al.
(författare)
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On 1-sum flows in undirected graphs
- 2016
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Ingår i: The Electronic Journal of Linear Algebra. - : INT LINEAR ALGEBRA SOC. - 1537-9582 .- 1081-3810. ; 31, s. 646-665
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Tidskriftsartikel (refereegranskat)abstract
- Let G = (V, E) be a simple undirected graph. For a given set L subset of R, a function omega: E -> L is called an L-flow. Given a vector gamma is an element of R-V , omega is a gamma-L-flow if for each v is an element of V, the sum of the values on the edges incident to v is gamma(v). If gamma(v) = c, for all v is an element of V, then the gamma-L-flow is called a c-sum L-flow. In this paper, the existence of gamma-L-flows for various choices of sets L of real numbers is studied, with an emphasis on 1-sum flows. Let L be a subset of real numbers containing 0 and denote L* := L \ {0}. Answering a question from [S. Akbari, M. Kano, and S. Zare. A generalization of 0-sum flows in graphs. Linear Algebra Appl., 438:3629-3634, 2013.], the bipartite graphs which admit a 1-sum R* -flow or a 1-sum Z* -flow are characterized. It is also shown that every k-regular graph, with k either odd or congruent to 2 modulo 4, admits a 1-sum {-1, 0, 1}-flow.
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| 3. |
- Saieed, Akbari, et al.
(författare)
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On linear transformations preserving at least one eigenvalue
- 2003
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Ingår i: Proceedings of the American Mathematical Society. - 0002-9939 .- 1088-6826. ; 132:6, s. 1621-1625
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Tidskriftsartikel (refereegranskat)abstract
- Let F be an algebraically closed field and T : Mn(F) −→ Mn(F)be a linear transformation. In this paper we show that if T preserves atleast one eigenvalue of each matrix, then T preserves all eigenvalues of eachmatrix. Moreover, for any infinite field F (not necessarily algebraically closed)we prove that if T : Mn(F) −→ Mn(F) is a linear transformation and for anyA ∈ Mn(F) with at least an eigenvalue in F, A and T(A) have at least onecommon eigenvalue in F, then T preserves the characteristic polynomial.
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