Geir Dahl

• Brualdi, Richard & Dahl, Geir (2025). The +/--rank of a (0,+/-)-matrix. Linear Algebra and its Applications. ISSN 0024-3795. 730, s. 318–343. doi: 10.1016/j.laa.2025.10.021. Fulltekst i vitenarkiv
• Brualdi, R.A. & Dahl, Geir (2025). Near-derangements and their polytopes. The Australasian Journal of Combinatorics. ISSN 1034-4942. 91(3), s. 366–386. Fulltekst i vitenarkiv Vis sammendrag

  A derangement is a permutation with no fixed point. We refer to permutations of {1, 2, . . . , n} with at most one fixed point as near-derangements, and study the corresponding set of permutation matrices Pn(≤1) . We determine a basis of the linear span of Pn(≤1) consisting of matrices in Pn(≤1). Also, we study the polytope determined by Pn(≤1). In addition, we investigate the polytope of all n × n doubly stochastic matrices with trace at most 1 and determine its extreme points, not all of which are permutation matrices.

• Brualdi, R.A. & Dahl, Geir (2024). Frobenius-König theorem for classes of (0,±1)-matrices. Discrete Mathematics. ISSN 0012-365X. 347(6). doi: 10.1016/j.disc.2024.113951. Fulltekst i vitenarkiv
• Brualdi, R.A. & Dahl, Geir (2024). Stirling permutations for partially ordered sets. Bulletin of the Institute of Combinatorics and its Applications. ISSN 1183-1278. 101, s. 114–127. Fulltekst i vitenarkiv Vis sammendrag

  We generalize the notion of a Stirling permutation of the multiset {1, 1, 2, 2, . . . , n, n} based on the usual linear order of the integers {1, 2, . . . , n} to any finite partially ordered set P, a P-Stirling permutation. We give an algorithmic characterization of P-Stirling permutations. A partially ordered set determines a transitive directed graph, and a further extension of Stirling permutations to directed graphs is discussed.

• Andrade, Enide & Dahl, Geir (2024). Combinatorial Fiedler theory and graph partition. Linear Algebra and its Applications. ISSN 0024-3795. 687, s. 229–251. doi: 10.1016/j.laa.2024.02.005. Fulltekst i vitenarkiv Vis sammendrag

  Partition problems in graphs are extremely important in applications, as shown in the Data Science and Machine Learning literature. One approach is spectral partitioning based on a Fiedler vector, i.e., an eigenvector corresponding to the second smallest eigenvalue a(G) of the Laplacian matrix LG of the graph G. This problem corresponds to the minimization of a quadratic form associated with LG, under certain constraints involving the ℓ2-norm. We introduce and investigate a similar problem, but using the ℓ1-norm to measure distances. This leads to a new parameter b(G) as the optimal value. We show that a well-known cut problem arises in this approach, namely the sparsest cut problem. We prove connectivity results and different bounds on this new parameter, relate to Fiedler theory and show explicit expressions for b(G) for trees. We also comment on an ℓ∞-norm version of the problem.

• Brualdi, R.A. & Dahl, Geir (2024). Multipermutations and Stirling Multipermutations. Graphs and Combinatorics. ISSN 0911-0119. 40(2). doi: 10.1007/s00373-024-02751-2. Fulltekst i vitenarkiv Vis sammendrag

  We consider multipermutations and a certain partial order, the weak Bruhat order, on this set. This generalizes the Bruhat order for permutations, and is defined in terms of containment of inversions. Different characterizations of this order are given. We also study special multipermutations called Stirling multipermutations and their properties.

• Brualdi, Richard A. & Dahl, Geir (2023). Multi-alternating sign matrices. The Electronic Journal of Linear Algebra. ISSN 1537-9582. 39, s. 261–282. doi: 10.13001/ela.2023.7471. Fulltekst i vitenarkiv Vis sammendrag

  We introduce a generalization of alternating sign matrices (ASMs) called multiASMs and develop some of their properties. Classes of multiASMs with specified row and column sum vectors R and S extend the classes of (0,1)-matrices with specified R and S. The special case when R=S is a constant vector, in particular all 2's, is treated in more detail. We also investigate the polytope spanned by a class of multiASMs. Finally, we discuss the possibility of defining a Bruhat order on a class of multiASMs.

• Dahl, Geir; Guterman, Alexander & Shteyner, Pavel (2023). Majorization for (0,±1)-matrices. Linear Algebra and its Applications. ISSN 0024-3795. 663, s. 200–221. doi: 10.1016/j.laa.2023.01.009. Fulltekst i vitenarkiv Vis sammendrag

  Matrix majorization is a generalization of the classical majorization for vectors. We study several basic questions concerning matrix majorization for (0;±1)-matrices, i.e., matrices whose entries are restricted to 0, 1 and -1. In particular, we characterize when the zero vector is weakly majorized by a matrix, and show related results. Connections to linear programming are discussed. We obtain simpler characterizations of majorization under different assumptions. Also, several results on directional and strong majorization for (0;±1)-matrices are shown.

• Brualdi, R.A. & Dahl, Geir (2023). Permutation graphs and the weak Bruhat order. The Art of Discrete and Applied Mathematics. ISSN 2590-9770. 6(3). doi: 10.26493/2590-9770.1536.8db. Fulltekst i vitenarkiv Vis sammendrag

  Permutation graphs are graphs associated with permutations where edges represent inversions. We study different classes of permutation graphs and isomorphic permutation graphs. A complete answer of a basic isomorphism question is given for trees. A connection to the majorization order for integer partitions is established, in connection with the notion of inversion vectors. Finally, we study special transitivity properties related to inversions of permutations.

• Andrade, Enide; Ciardo, Lorenzo & Dahl, Geir (2022). Perron values and classes of trees. Linear Algebra and its Applications. ISSN 0024-3795. 639, s. 135–158. doi: 10.1016/j.laa.2022.01.005. Fulltekst i vitenarkiv Vis sammendrag

  The bottleneck matrix M of a rooted tree T is a combinatorial object encoding the spatial distribution of the vertices with respect to the root. The spectral radius of M, known as the Perron value of the rooted tree, is closely related to the theory of the algebraic connectivity. In this paper, we investigate the Perron values of various classes of rooted trees by making use of combinatorial and linear-algebraic techniques. This results in multiple bounds on the Perron values of these classes, which can be straightforwardly applied to provide information on the algebraic connectivity.

• Brualdi, Richard A. & Dahl, Geir (2021). Sign-restricted matrices of 0's, 1's, and −1's. Linear Algebra and its Applications. ISSN 0024-3795. 615, s. 77–103. doi: 10.1016/j.laa.2021.01.001. Fulltekst i vitenarkiv Vis sammendrag

  We study sign-restricted matrices (SRMs), a class of rectangular (0, ±1)-matrices generalizing the alternating sign matrices (ASMs). In an SRM each partial column sum, starting from row 1, equals 0 or 1, and each partial row sum, starting from column 1, is nonnegative. We determine the maximum number of nonzeros in SRMs and characterize the possible row and column sum vectors. Moreover, a number of results on interchange operations are shown, both for SRMs and, more generally, for (0, ±1)-matrices. The Bruhat order on ASMs can be extended to SRMs with the result a distributive lattice. Also, we study polytopes associated with SRMs and some relates decompositions.

• Brualdi, R.A. & Dahl, Geir (2021). Diagonal Sums of Doubly Stochastic Matrices. Linear and multilinear algebra. ISSN 0308-1087. doi: 10.1080/03081087.2021.1901844. Fulltekst i vitenarkiv
• Brualdi, R.A. & Dahl, Geir (2021). Convex (0,1)-Matrices and Their Epitopes, Discrete Applied Mathematics. ISSN 0166-218X. 297, s. 21–34. doi: 10.1016/j.dam.2021.02.038. Fulltekst i vitenarkiv
• Brualdi, R.A. & Dahl, Geir (2021). Alternating Sign and Sign-Restricted Matrices: Representations and Partial Orders. The Electronic Journal of Linear Algebra. ISSN 1537-9582. 37. doi: 10.13001/ela.2021.6513. Fulltekst i vitenarkiv Vis sammendrag

  Sign-restricted matrices (SRMs) are (0,±1)-matrices where, ignoring 0's, the signs in each column alternate beginning with a +1 and all partial row sums are nonnegative. The most investigated of these matrices are the alternating sign matrices (ASMs), where the rows also have the alternating sign property, and all row and column sums equal 1. We introduce monotone triangles to represent SRMs and investigate some of their properties and connections to certain polytopes. We also investigate two partial orders for ASMs related to their patterns alternating cycles and show a number of combinatorial properties of these orders.

• Brualdi, R.A. & Dahl, Geir (2020). Permutation matrices, their discrete derivatives and extremal properties. Vietnam Journal of Mathematics. ISSN 2305-221X. 48, s. 719–740. doi: 10.1007/s10013-020-00392-5. Fulltekst i vitenarkiv Vis sammendrag

  Abstract For a permutation π , and the corresponding permutation matrix, we introduce the notion of discrete derivative , obtained by taking differences of successive entries in π . We characterize the possible derivatives of permutations, and consider questions for permutations with certain properties satisfied by the derivative. For instance, we consider permutations with distinct derivatives, and the relationship to so-called Costas arrays.

• Ciardo, Lorenzo; Dahl, Geir & Kirkland, Steve (2020). On Kemeny's constant for trees with fixed order and diameter. Linear and multilinear algebra. ISSN 0308-1087. doi: 10.1080/03081087.2020.1796905. Fulltekst i vitenarkiv
• Brualdi, R.A. & Dahl, Geir (2020). Alternating Sign Matrices: Extensions, König-properties, and Primary Sum-Sequences. Graphs and Combinatorics. ISSN 0911-0119. 36, s. 63–92. doi: 10.1007/s00373-019-02119-x. Fulltekst i vitenarkiv Vis sammendrag

  This paper is concerned with properties of permutation matrices and alternating sign matrices (ASMs). An ASM is a square (0,±1)-matrix such that, ignoring 0’s, the 1’s and −1’s in each row and column alternate, beginning and ending with a 1. We study extensions of permutation matrices into ASMs by changing some zeros to +1 or −1. Furthermore, several properties concerning the term rank and line covering of ASMs are shown. An ASM A is determined by a sum-matrix Σ(A) whose entries are the sums of the entries of its leading submatrices (so determined by the entries of A). We show that those sums corresponding to the nonzero entries of a permutation matrix determine all the entries of the sum-matrix and investigate some of the properties of the resulting sequence of numbers. Finally, we investigate the lattice-properties of the set of ASMs (of order n), where the partial order comes from the Bruhat order for permutation matrices.

• Andrade, Enide; Dahl, Geir; Leal, Laura & Robbiano, Maria (2019). New bounds for the signless Laplacian spread. Linear Algebra and its Applications. ISSN 0024-3795. 566, s. 98–120. doi: 10.1016/j.laa.2018.12.019. Fulltekst i vitenarkiv Vis sammendrag

  Let G be an undirected simple graph. The signless Laplacian spread of G is defined as the maximum distance of pairs of its signless Laplacian eigenvalues. This paper establishes some new bounds, both lower and upper, for the signless Laplacian spread. Several of these bounds depend on invariant parameters of the graph. We also use a minmax principle to find several lower bounds for this spectral invariant.

• Dahl, Geir; Guterman, Alexander & Shteyner, Pavel (2019). Majorization for (0,1)-matrices. Linear Algebra and its Applications. ISSN 0024-3795. 585, s. 147–163. doi: 10.1016/j.laa.2019.09.038. Fulltekst i vitenarkiv
• Andrade, Enide; Ciardo, Lorenzo & Dahl, Geir (2019). Combinatorial Perron Parameters for Trees. Linear Algebra and its Applications. ISSN 0024-3795. 566, s. 138–166. doi: 10.1016/j.laa.2018.12.028. Fulltekst i vitenarkiv
• Dahl, Geir; Guterman, Alexander & Shteyner, Pavel (2018). Majorization for matrix classes. Linear Algebra and its Applications. ISSN 0024-3795. 555, s. 201–221. doi: 10.1016/j.laa.2018.06.003. Fulltekst i vitenarkiv Vis sammendrag

  We introduce a new majorization order for classes (sets) of matrices which generalizes several existing notions of matrix majorization. Roughly, the notion says that every matrix in one class is majorized by some matrix in the other class. The motivation to study this majorization concept comes from mathematical statistics and involves the information content in experiments. This connection is briefly described. We investigate properties of this new order both of algebraic and geometric character. In particular, we establish results on so-called minimal cover classes with respect to the introduced majorization.

• Brualdi, R.A. & Dahl, Geir (2018). Alternating sign matrices, related (0,1)-matrices, and the Smith normal form. Linear Algebra and its Applications. ISSN 0024-3795. s. 1–19. doi: 10.1016/j.laa.2018.03.027. Fulltekst i vitenarkiv Vis sammendrag

  We investigate the Smith Normal Form (SNF) of alternating sign matrices (ASMs) and related matrices of 0’s and 1’s ((0, 1)-matrices). We identify certain classes of ASMs and (0, 1)-matrices whose SNFs are (0, 1)-matrices. We relate some of our work to various ranks, in particular, the (0, 1)-rank of a (0, 1)-matrix, that is, the bipartite partition number of a bipartite graph.

• Brualdi, R.A. & Dahl, Geir (2018). The interval structure of (0,1)-matrices. Discrete Applied Mathematics. ISSN 0166-218X. doi: 10.1016/j.dam.2018.03.024. Fulltekst i vitenarkiv Vis sammendrag

  Let A be an n × n (0, ∗)-matrix, so each entry is 0 or ∗. An A-interval matrix is a (0, 1)-matrix obtained from A by choosing some ∗’s so that in every interval of consecutive ∗’s, in a row or column of A, exactly one ∗ is chosen and replaced with a 1, and every other ∗ is replaced with a 0. We consider the existence questions for A-interval matrices, both in general, and for specific classes of such A defined by permutation matrices. Moreover, we discuss uniqueness and the number of A-permutation matrices, as well as properties of an associated graph.

• Brualdi, R.A. & Dahl, Geir (2018). Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Squares. Advances in Applied Mathematics. ISSN 0196-8858. 95, s. 116–151. doi: 10.1016/j.aam.2017.11.005. Fulltekst i vitenarkiv Vis sammendrag

  An alternating sign matrix, or ASM, is a (0, ±1)-matrix where the nonzero entries in each row and column alternate in sign. We generalize this notion to hypermatrices: an n × n × n hypermatrix A = [aijk] is an alternating sign hypermatrix, or ASHM, if each of its planes, obtained by fixing one of the three indices, is an ASM. Several results concerning ASHMs are shown, such as finding the maximum number of nonzeros of an n × n × n ASHM, and properties related to Latin squares. Moreover, we investigate completion problems, in which one asks if a subhypermatrix can be completed (extended) into an ASHM. We show several theorems of this type.

• Brualdi, R.A. & Dahl, Geir (2017). Alternating sign matrices, extensions and related cones. Advances in Applied Mathematics. ISSN 0196-8858. 86, s. 19–49. doi: 10.1016/j.aam.2016.12.001. Fulltekst i vitenarkiv Vis sammendrag

  An alternating sign matrix , or ASM, is a (0,±1)(0,±1)-matrix where the nonzero entries in each row and column alternate in sign, and where each row and column sum is 1. We study the convex cone generated by ASMs of order n , called the ASM cone, as well as several related cones and polytopes. Some decomposition results are shown, and we find a minimal Hilbert basis of the ASM cone. The notion of (±1)(±1)-doubly stochastic matrices and a generalization of ASMs are introduced and various properties are shown. For instance, we give a new short proof of the linear characterization of the ASM polytope, in fact for a more general polytope. Finally, we investigate faces of the ASM polytope, in particular edges associated with permutation matrices.

• Agra, Agostinho; Dahl, Geir; Haufmann, Torkel Andreas & Pinheiro, Sofia (2017). The k-regular induced subgraph problem. Discrete Applied Mathematics. ISSN 0166-218X. 222, s. 14–30. doi: 10.1016/j.dam.2017.01.029. Fulltekst i vitenarkiv Vis sammendrag

  We consider the problem of finding a maximum k-regular induced subgraph of a graph G. Theoretical results are established to compare upper bounds obtained from different techniques, including bounds from quadratic programming, Lagrangian relaxation and integer programming. This general problem includes well-known subproblems as particular cases of k. In this paper we focus on two particular cases. The case k=1k=1 which is the maximal cardinality strong-matching and the case of finding the maximal cardinality family of induced cycles (k=2k=2). For each one of the two cases, combinatorial algorithms are presented to solve the problem when graphs have particular structures and polyhedral descriptions of the convex hull of the corresponding feasible set are given. Computational tests are reported to compare the different upper bounds with the optimal values for different values of k, and to test the effectiveness of the inequalities introduced.

• Andrade, Enide & Dahl, Geir (2016). Combinatorial Perron values of trees and bottleneck matrices. Linear and multilinear algebra. ISSN 0308-1087. 65(12), s. 2387–2405. doi: 10.1080/03081087.2016.1274363. Fulltekst i vitenarkiv Vis sammendrag

  The algebraic connectivity a(G) of a graph G is an important parameter, defined as the second smallest eigenvalue of the Laplacian matrix of G. If T is a tree, a(T) is closely related to the Perron values (spectral radius) of so-called bottleneck matrices of subtrees of T. In this setting we introduce a new parameter called the combinatorial Perron value ρc. This value is a lower bound on the Perron value of such subtrees; typically ρc is a good approximation to ρ. We compute exact values of ρc for certain special subtrees. Moreover, some results concerning ρc when the tree is modified are established, and it is shown that, among trees with given distance vector (from the root), ρc is maximized for caterpillars. This research was first published in Linear and Multilinear Algebra. © Taylor & Francis.

• Brualdi, Richard A.; Dahl, Geir & Fritscher, Eliseu (2016). Doubly Stochastic Matrices and the Bruhat Order. Czechoslovak Mathematical Journal. ISSN 0011-4642. 66(3), s. 681–700. doi: 10.1007/s10587-016-0286-6. Fulltekst i vitenarkiv Vis sammendrag

  The Bruhat order is defined in terms of an interchange operation on the set of permutation matrices of order n which corresponds to the transposition of a pair of elements in a permutation. We introduce an extension of this partial order, which we call the stochastic Bruhat order, for the larger class Ωn of doubly stochastic matrices (convex hull of n×n permutation matrices). An alternative description of this partial order is given. We define a class of special faces of Ωn induced by permutation matrices, which we call Bruhat faces. Several examples of Bruhat faces are given and several results are presented. This research was first published in Czechoslovak Mathematical Journal. © Springer Verlag.

• Dahl, Geir & Haufmann, Torkel Andreas (2016). Zero-one completely positive matrices and the A(R, S) classes. Special Matrices. ISSN 2300-7451. 4(1), s. 296–304. doi: 10.1515/spma-2016-0024. Fulltekst i vitenarkiv Vis sammendrag

  A matrix of the form A = BBT where B is nonnegative is called completely positive (CP). Berman and Xu (2005) investigated a subclass of CP-matrices, called f0, 1g-completely positive matrices. We introduce a related concept and show connections between the two notions. An important relation to the so-called cut cone is established. Some results are shown for f0, 1g-completely positive matrices with given graphs, and for {0,1}-completely positive matrices constructed from the classes of (0, 1)-matrices with fixed row and column sums.

• Barragan, Luis A.; Dahl, Geir; Dominguez, Alberto & Otin, Arantxa (2016). Zero sum sign-central matrices and applications. Linear Algebra and its Applications. ISSN 0024-3795. 505, s. 109–125. doi: 10.1016/j.laa.2016.04.034. Fulltekst i vitenarkiv Vis sammendrag

  A matrix with a nonzero nonnegative vector in its null space is called central . We study classes of central matrices having zero column sums. The study is motivated by an engineering application concerning induction heating where central matrices provide a way to control the energy flow over time. A (±1)(±1)-matrix A is called a ZSC-matrix (zero sum sign-central) if each matrix with the same sign pattern as A and having zero column sums is central. We establish several classes of ZSC-matrices, and give separate sufficient and necessary conditions for a matrix to be ZSC. Moreover, we give algorithms for finding central matrices that are used for power control in induction heating, and illustrate these by some numerical examples.

• Dahl, Geir (2015). L-rays of permutation matrices and doubly stochastic matrices. Linear Algebra and its Applications. ISSN 0024-3795. 480, s. 127–143. doi: 10.1016/j.laa.2015.04.033. Fulltekst i vitenarkiv
• Brualdi, Richard A. & Dahl, Geir (2015). Strict Sign-Central Matrices. SIAM Journal on Matrix Analysis and Applications. ISSN 0895-4798. 36(3), s. 1202–1220. doi: 10.1137/140995428. Fulltekst i vitenarkiv
• Abreu, Nair; Costa, Liliana; Dahl, Geir & Martins, Enide (2014). The skeleton of acyclic Birkhoff polytopes. Linear Algebra and its Applications. ISSN 0024-3795. 457, s. 29–48. doi: 10.1016/j.laa.2014.05.021. Fulltekst i vitenarkiv
• Dahl, Geir (2014). The Laplacian energy of threshold graphs and majorization. Linear Algebra and its Applications. ISSN 0024-3795. 469, s. 518–530. doi: 10.1016/j.laa.2014.12.002. Fulltekst i vitenarkiv
• Charina, Maria & Dahl, Geir (2014). Subdivision schemes, network flows and linear optimization. Advances in Computational Mathematics. ISSN 1019-7168. 41, s. 507–528. doi: 10.1007/s10444-014-9371-2. Fulltekst i vitenarkiv
• Brualdi, Richard A. & Dahl, Geir (2013). An extension of the polytope of doubly stochastic matrices. Linear and multilinear algebra. ISSN 0308-1087. 61(3), s. 393–408. doi: 10.1080/03081087.2012.689980. Fulltekst i vitenarkiv Vis sammendrag

  This is an Author's Accepted Manuscript of an article published in Linear and Multilinear Algebra Volume 61, Issue 3, 2013. Published online: 07 Jun 2012. Copyright Taylor & Francis, available online

• Dahl, Geir (2013). A note on majorization transforms and Ryser’s algorithm. Special Matrices. ISSN 2300-7451. 1, s. 17–24. doi: 10.2478/spma-2013-0004. Fulltekst i vitenarkiv
• Brualdi, Richard A. & Dahl, Geir (2013). Majorization for partially ordered sets. Discrete Mathematics. ISSN 0012-365X. 313(22), s. 2592–2601. doi: 10.1016/j.disc.2013.08.003. Fulltekst i vitenarkiv
• Dahl, Geir & Zhang, Fuzhen (2013). Integral majorization polytopes. Discrete Mathematics, Algorithms and Applications (DMAA). ISSN 1793-8309. 5(3). doi: 10.1142/s1793830913500195. Fulltekst i vitenarkiv
• Dahl, Geir (2012). Martingale matrix classes and polytopes. Linear Algebra and its Applications. ISSN 0024-3795. 437(7), s. 1722–1733. doi: 10.1016/j.laa.2012.04.042. Fulltekst i vitenarkiv
• Dahl, Geir (2012). A matrix-based ranking method with application to tennis. Linear Algebra and its Applications. ISSN 0024-3795. 437(1), s. 26–36. doi: 10.1016/j.laa.2012.02.002. Fulltekst i vitenarkiv
• Benth, Fred Espen; Dahl, Geir & Mannino, Carlo (2012). Computing Optimal Recovery Policies for Financial Markets. Operations Research. ISSN 0030-364X. 60(6), s. 1373–1388. doi: 10.1287/opre.1120.1112. Fulltekst i vitenarkiv
• Brualdi, R.A. & Dahl, Geir (2012). Majorization classes of integral matrices. Linear Algebra and its Applications. ISSN 0024-3795. 436(4), s. 802–813. doi: 10.1016/j.laa.2010.11.036. Fulltekst i vitenarkiv
• Dahl, Geir (2011). Polytopes related to interval vectors and incidence matrices. Linear Algebra and its Applications. ISSN 0024-3795. 435(11), s. 2955–2960. doi: 10.1016/j.laa.2011.05.026. Fulltekst i vitenarkiv
• Dahl, Geir (2010). Majorization permutahedra and (0,1)-matrices. Linear Algebra and its Applications. ISSN 0024-3795. 432(12), s. 3265–3271. doi: 10.1016/j.laa.2010.01.024. Fulltekst i vitenarkiv

Se alle arbeider i NVA

• Dahl, Geir & Brualdi, Richard A. (2026). A new combinatorial rank concept and its challenges. Fulltekst i vitenarkiv
• Brualdi, R.A. & Dahl, Geir (2025). Near-derangements and their polytopes. Fulltekst i vitenarkiv
• Dahl, Geir & Brualdi, R.A. (2024). Combinatorial Matrices and Frobenius Theory. Fulltekst i vitenarkiv
• Dahl, Geir; Brualdi, R.A. & Andrade, Enide (2024). Frobenius, Combinatorial Matrices and Graph Partition. Fulltekst i vitenarkiv
• Dahl, Geir (2024). Combinatorial Matrix Theory - an invitation. Fulltekst i vitenarkiv
• Dahl, Geir & Brualdi, R.A. (2024). Combinatorial Matrices and Frobenius Theory. Fulltekst i vitenarkiv
• Andrade, Enide & Dahl, Geir (2023). New results on graph partition and Fiedler theory. Fulltekst i vitenarkiv
• Dahl, Geir & Brualdi, R.A. (2023). Alternating Sign Matrices and Generalizations. Fulltekst i vitenarkiv
• Guterman, Alexander; Dahl, Geir & Shteyner, Pavel (2023). On combinatorial matrix majorizations. Fulltekst i vitenarkiv
• Dahl, Geir; Guterman, Alexander & Shteyner, Pavel (2023). Majorization orders for (0,+1, -1)-matrices. Fulltekst i vitenarkiv
• Dahl, Geir & Brualdi, R.A. (2022). The permutation and alternating sign matrix cones. Fulltekst i vitenarkiv
• Dahl, Geir & Brualdi, R.A. (2022). Combinatorial matrix cones. Fulltekst i vitenarkiv
• Dahl, Geir (2021). Kombinatorikk, geometri og Munch! Fulltekst i vitenarkiv
• Brualdi, R.A. & Dahl, Geir (2021). Diagonal sums of doubly stochastic matrices. Fulltekst i vitenarkiv
• Dahl, Geir; Munthe-Kaas, Antonella Zanna; Rønquist, Einar; Rypdal, Martin Wibe; Erfjord, Ingvald & Strømskag, Heidi [Vis alle 10 forfattere av denne artikkelen] (2021). Eksamen i matematikk er på ville veier. Aftenposten (morgenutg. : trykt utg.). ISSN 0804-3116. Fulltekst i vitenarkiv
• Dahl, Geir (2021). Doubly stochastic matrices: old and new results. Fulltekst i vitenarkiv
• Andrade, Enide; Dahl, Geir; Leal, Laura & Robbiano, Maria (2019). New bounds for the signless Laplacian spread. Fulltekst i vitenarkiv
• Brualdi, R.A. & Dahl, Geir (2019). Matrix covering and A-interval matrices. Fulltekst i vitenarkiv
• Dahl, Geir (2019). Majorization - some recent developments in matrix theory. Fulltekst i vitenarkiv
• Dahl, Geir; Brualdi, R.A.; Guterman, Alexander & Shteyner, Pavel (2019). Matrix covering problems. Fulltekst i vitenarkiv
• Dahl, Geir; Andrade, Enide & Ciardo, Lorenzo (2018). Combinatorial Perron Parameters and Trees. Fulltekst i vitenarkiv
• Dahl, Geir & Brualdi, R.A. (2018). Alternating Sign Matrices, Polyhedra and Majorization. Fulltekst i vitenarkiv
• Brualdi, R.A. & Dahl, Geir (2017). Alternating sign matrices and related polyhedra. Fulltekst i vitenarkiv
• Dahl, Geir (2017). Laplacian energy, threshold graphs and majorization. Fulltekst i vitenarkiv
• Brualdi, R.A. & Dahl, Geir (2017). Alternating sign matrices and polyhedra. Fulltekst i vitenarkiv
• Dahl, Geir; Ranestad, Kristian & Hole, Arne (2017). Programmering rammer dybdelæring i matematikk. s. 22–23. Fulltekst i vitenarkiv
• Andrade, Enide & Dahl, Geir (2016). Combinatorial Perron values. Fulltekst i vitenarkiv
• Dahl, Geir & Brualdi, R.A. (2016). Doubly stochastic matrices and the Bruhat order. Fulltekst i vitenarkiv
• Dahl, Geir & Haufmann, Torkel Andreas (2016). Zero-one completely positive matrices and the A(R,S )matrix classes. Fulltekst i vitenarkiv
• Andrade, Enide & Dahl, Geir (2016). Combinatorial Perron values. Fulltekst i vitenarkiv
• Haufmann, Torkel Andreas; Agra, Agostinho; Dahl, Geir & Pinheiro, Sofia (2016). The k-regular induced subgraph problem. Fulltekst i vitenarkiv
• Dahl, Geir (2016). Some Topics in Combinatorial Matrix Theory. Fulltekst i vitenarkiv
• Agra, Agostinho; Dahl, Geir; Haufmann, Torkel Andreas & Pinheiro, Sofia (2015). The k-regular induced subgraph problem for k=1,2. Fulltekst i vitenarkiv
• Dahl, Geir (2015). Permutation matrices, doubly stochastic matrices and their L– rays. Fulltekst i vitenarkiv
• Dahl, Geir (2015). Combinatorial Matrix Theory and Majorization. Fulltekst i vitenarkiv
• Dahl, Geir (2015). Majorization, (0,1)-matrices and Laplacian energy. Fulltekst i vitenarkiv
• Agra, Agostinho; Dahl, Geir; Haufmann, Torkel Andreas & Pinheiro, Sofia (2015). Maximal induced k-regular subgraphs. Fulltekst i vitenarkiv
• Haufmann, Torkel Andreas & Dahl, Geir (2014). Optimization and classes of completely positive matrices. Fulltekst i vitenarkiv
• Dahl, Geir & Brualdi, R.A. (2014). Majorization for partially ordered sets. Fulltekst i vitenarkiv
• Dahl, Geir (2014). Traveling Salesman Problem: En matematisk rundtur. Fulltekst i vitenarkiv
• Dahl, Geir (2014). Majorization and combinatorial matrix theory. Fulltekst i vitenarkiv
• Dahl, Geir (2013). Nytt Munch museum: et SKRIK etter kombinatorikk og geometri!! Fulltekst i vitenarkiv
• Dahl, Geir (2013). Majorization transforms and Ryser’s algorithm. Fulltekst i vitenarkiv
• Dahl, Geir (2013). An introduction to Majorization: theory and applications. Fulltekst i vitenarkiv
• Dahl, Geir (2012). A taste of Szemerédi's combinatorics. Fulltekst i vitenarkiv
• Bebiano, Natália; Dahl, Geir; Silva, José Dias da & Fonseca, Carlos M. da (2012). Preface to the Proceedings of the Coimbra Meeting on 0-1 Matrix Theory and Related Topics. Linear Algebra and its Applications. ISSN 0024-3795. 436(4), s. 789–790. doi: 10.1016/j.laa.2011.09.028. Fulltekst i vitenarkiv
• Dahl, Geir (2012). Majorization, matrix polytopes and combinatorics. Fulltekst i vitenarkiv
• Dahl, Geir (2012). Lineær algebra, Google og rangering av websider. Fulltekst i vitenarkiv
• Dahl, Geir (2012). Google: hvordan man blir styrtrik på en matematisk likning! Fulltekst i vitenarkiv
• Dahl, Geir & Brualdi, R.A. (2012). Generalized Birkhoff polytopes and majorization. Fulltekst i vitenarkiv
• Brualdi, R.A. & Dahl, Geir (2012). An extension of the polytope of doubly stochastic matrices. Fulltekst i vitenarkiv
• Dahl, Geir (2011). Kombinatorikk på roterommet! Fulltekst i vitenarkiv
• Dahl, Geir (2011). Majorization in matrix theory: a brief introduction and some recent progress. Fulltekst i vitenarkiv
• Dahl, Geir (2011). Martingale matrix classes. Fulltekst i vitenarkiv
• Dahl, Geir & Dahl, Kristina Rognlien (2012). Linear optimization and mathematical finance. Universitetet i Oslo. Fulltekst i vitenarkiv

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