Biased graphs IV: Geometrical realizations

DOI10.1016/S0095-8956(03)00035-2zbMath1031.05034MaRDI QIDQ1417965

Thomas Zaslavsky

Publication date: 6 January 2004

Published in: Journal of Combinatorial Theory. Series B (Search for Journal in Brave)

zbMATH Keywords

Poincaré polynomialrepresentationcharacteristic polynomialChromatic polynomialBiased graphMatroidMenelausCevaWhitney-number polynomialArrangement of hyperplanesBalanceBias matroidGain graphGeometric semilatticeGraphic lift matroidHyperplane separationLogic of equationsOrthographic representationPermutation gain graphPermutation representationPythagorean representationThick biased graphWhitney 2-isomorphism

Mathematics Subject Classification ID

Exact enumeration problems, generating functions (05A15) Graphs and linear algebra (matrices, eigenvalues, etc.) (05C50) Arrangements of points, flats, hyperplanes (aspects of discrete geometry) (52C35) Combinatorial aspects of matroids and geometric lattices (05B35) Incidence structures embeddable into projective geometries (51A45) Signed and weighted graphs (05C22) General logic (03B99)

Related Items (33)

Matroids arising from electrical networks ⋮ Matrix representations of frame and lifted-graphic matroids correspond to gain functions ⋮ Bicircular matroids representable over \(\mathrm{GF}(4)\) or \(\mathrm{GF}(5)\) ⋮ Matroids of gain graphs in applied discrete geometry ⋮ Biased graphs. VII: Contrabalance and antivoltages ⋮ Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements ⋮ Minimal non-orientable matroids in a projective plane ⋮ The number of nowhere-zero flows on graphs and signed graphs ⋮ A Generalisation of the Matroid Lift Construction ⋮ On the adjacency matrix of a complex unit gain graph ⋮ When does a biased graph come from a group labelling? ⋮ On the determinant of the Laplacian matrix of a complex unit gain graph ⋮ Almost balanced biased graph representations of frame matroids ⋮ On the structure of matroids arising from the gain graphs ⋮ Spectral properties of complex unit gain graphs ⋮ Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species ⋮ Modular construction of free hyperplane arrangements ⋮ A reciprocal eigenvalue property for unicyclic weighted directed graphs with weights from \(\{\pm 1,{\pm}i\}\) ⋮ Biased graphs. II: The three matroids ⋮ Gain-line graphs via \(G\)-phases and group representations ⋮ Bijections between affine hyperplane arrangements and valued graphs ⋮ Laplacian matrices of general complex weighted directed graphs ⋮ Yes, the “missing axiom” of matroid theory is lost forever ⋮ On excluded minors for classes of graphical matroids ⋮ Lindström's conjecture on a class of algebraically non-representable matroids ⋮ Infinitely many excluded minors for frame matroids and for lifted-graphic matroids ⋮ On excluded minors of connectivity 2 for the class of frame matroids ⋮ Oriented gain graphs, line graphs and eigenvalues ⋮ A group representation approach to balance of gain graphs ⋮ On the Falk invariant of Shi and linial arrangements ⋮ Describing quasi-graphic matroids ⋮ Biased graphs. VI: Synthetic geometry ⋮ The graphs that have antivoltages using groups of small order

Cites Work

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• Biased graphs. I: Bias, balance, and gains
• The biased graphs whose matroids are binary
• Combinatorics and topology of complements of hyperplanes
• Convexity in oriented matroids
• Signed graphs
• Chromatic invariants of signed graphs
• Orientation of signed graphs
• Orientability of matroids
• Frame matroids and biased graphs
• Unavoidable minors of large 3-connected matroids
• A mathematical bibliography of signed and gain graphs and allied areas
• \(l\)-matrices and a characterization of binary matroids
• Perpendicular dissections of space
• Biased graphs whose matroids are special binary matroids
• Supersolvable frame-matroid and graphic-lift lattices
• Branch-width and well-quasi-ordering in matroids and graphs.
• On inequivalent representations of matroids over finite fields
• A class of geometric lattices based on finite groups
• Characteristic polynomials of subspace arrangements and finite fields
• On the notion of balance of a signed graph
• On the Interpretation of Whitney Numbers Through Arrangements of Hyperplanes, Zonotopes, Non-Radon Partitions, and Orientations of Graphs
• BICIRCULAR GEOMETRY AND THE LATTICE OF FORESTS OF A GRAPH
• Facing up to arrangements: face-count formulas for partitions of space by hyperplanes
• Lectures on matroids
• A certain generalization of the theorems of Menelaos and Ceva

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