Bounding the coefficients of the characteristic polynomials of simple binary matroids

AbstractWe give an upper bound and a class of lower bounds on the coefficients of the characteristic polynomial of a simple binary matroid. This generalizes the corresponding bounds for graphic matroids of Li and Tian (1978) [3], as well as a matroid lower bound of Björner (1980) [1] for simple binary matroids. As the flow polynomial of a graph G is the characteristic polynomial of the dual matroid M∗(G), the bound applies to flow polynomials

On the Structure of 3-connected Matroids and Graphs

An element e of a 3-connected matroid M is essential if neither the deletion M\e nor the contraction M/e is 3-connected. Tutte\u27s Wheels and Whirls Theorem proves that the only 3-connected matroids in which every element is essential are the wheels and whirls. In this paper, we consider those 3-connected matroids that have some non-essential elements, showing that every such matroid M must have at least two such elements. We prove that the essential elements of M can be partitioned into classes where two elements are in the same class if M has a fan, a maximal partial wheel, containing both. We also prove that if an essential element e of M is in more than one fan, then that fan has three or five elements; in the latter case, e is in exactly three fans. Moreover, we show that if M has a fan with 2k or 2k + 1 elements for some k ≥ 2, then M can be obtained by sticking together a (k + 1)-spoked wheel and a certain 3-connected minor of M. The results proved here will be used elsewhere to completely determine all 3-connected matroids with exactly two non-essential elements. © 2000 Academic Press

A Generalization of a Theorem of Dirac

AbstractIn this paper, we give a generalization of a well-known result of Dirac that given any k vertices in a k-connected graph where k⩾2, there is a circuit containing all of them. We also generalize a result of Häggkvist and Thomassen. Our main result partially answers an open matroid question of Oxley

On matroid connectivity

If M is a loopless matroid in which M/vbX and M/vbY are connected and X∩Y is non-empty, then one easily shows that M/vb(X∪Y) is connected. Likewise, it is straightforward to show that if G and H are n-connected graphs having at least n common vertices, then G ∪ H is n-connected. The purpose of this note is to prove a matroid connectivity result that is a common generalization of these two observations. © 1995

Beta Invariant and Chromatic Uniqueness of Wheels

A graph G is chromatically unique if its chromatic polynomial completely determines the graph. An n-spoked wheel, Wn, is shown to be chromatically unique when n ≥ 4 is even [S.-J. Xu and N.-Z. Li, The chromaticity of wheels, Discrete Math. 51 (1984) 207–212]. When n is odd, this problem is still open for n ≥ 15 since 1984, although it was shown by di erent researchers that the answer is no for n = 5, 7, yes for n = 3, 9, 11, 13, and unknown for other odd n. We use the beta invariant of matroids to prove that if M is a 3-connected matroid such that |E(M)| = |E(Wn)| and β (M) = β (M(Wn)), where β (M) is the beta invariant of M, then M ≅ M(Wn). As a consequence, if G is a 3-connected graph such that the chromatic (or flow) polynomial of G equals to the chromatic (or flow) polynomial of a wheel, then G is isomorphic to the wheel. The examples for n = 3, 5 show that the 3-connectedness condition may not be dropped. We also give a splitting formula for computing the beta invariants of general parallel connection of two matroids as well as the 3-sum of two binary matroids. This generalizes the corresponding result of Brylawski [A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154 (1971) 1–22]

Characterizing binary matroids with no P9-minor

In this paper, we give a complete characterization of binary matroids with no P9-minor. A 3-connected binary matroid M has no P9-minor if and only if M is a 3-connected regular matroid, a binary spike with rank at least four, one of the internally 4-connected non-regular minors of a special 16-element matroid Y16, or a matroid obtained by 3-summing copies of the Fano matroid to a 3-connected cographic matroid M∗(K3,n), M∗(K3,n\u27), M∗(K3,n\u27\u27), or M∗(K3,n\u27\u27\u27) (n ≥ 2). Here the simple graphs K3,n\u27, K3,n\u27\u27, and K3,n\u27\u27\u27 are obtained from K3,n by adding one, two, or three edges in the color class of size three, respectively

On Tutte polynomial uniqueness of twisted wheels

AbstractA graph G is called T-unique if any other graph having the same Tutte polynomial as G is isomorphic to G. Recently, there has been much interest in determining T-unique graphs and matroids. For example, de Mier and Noy [A. de Mier, M. Noy, On graphs determined by their Tutte polynomials, Graphs Combin. 20 (2004) 105–119; A. de Mier, M. Noy, Tutte uniqueness of line graphs, Discrete Math. 301 (2005) 57–65] showed that wheels, ladders, Möbius ladders, square of cycles, hypercubes, and certain class of line graphs are all T-unique. In this paper, we prove that the twisted wheels are also T-unique

Composite Biased Rotations for Precise Raman Control of Spinor Matterwaves

Precise control of hyperfine matterwaves via Raman excitations is instrumental to a class of atom-based quantum technology. We investigate the Raman spinor control technique for alkaline atoms in an intermediate regime of single-photon detuning where a choice can be made to balance the Raman excitation power efficiency with the control speed, excited-state adiabatic elimination, and spontaneous emission suppression requirements. Within the regime, rotations of atomic spinors by the Raman coupling are biased by substantial light shifts. Taking advantage of the fixed bias angle, we show that composite biased rotations can be optimized to enable precise ensemble spinor matterwave control within nanoseconds, even for multiple Zeeman pseudo-spins defined on the hyperfine ground states and when the laser illumination is strongly inhomogeneous. Our scheme fills a technical gap in light pulse atom interferometry, for achieving high speed Raman spinor matterwave control with moderate laser power.Comment: 11 pages, 6 figure