5 research outputs found
The adjacency matroid of a graph
If is a looped graph, then its adjacency matrix represents a binary
matroid on . may be obtained from the delta-matroid
represented by the adjacency matrix of , but is less sensitive to
the structure of . Jaeger proved that every binary matroid is for
some [Ann. Discrete Math. 17 (1983), 371-376].
The relationship between the matroidal structure of and the
graphical structure of has many interesting features. For instance, the
matroid minors and are both of the form
where may be obtained from using local
complementation. In addition, matroidal considerations lead to a principal
vertex tripartition, distinct from the principal edge tripartition of
Rosenstiehl and Read [Ann. Discrete Math. 3 (1978), 195-226]. Several of these
results are given two very different proofs, the first involving linear algebra
and the second involving set systems or delta-matroids. Also, the Tutte
polynomials of the adjacency matroids of and its full subgraphs are closely
connected to the interlace polynomial of Arratia, Bollob\'{a}s and Sorkin
[Combinatorica 24 (2004), 567-584].Comment: v1: 19 pages, 1 figure. v2: 20 pages, 1 figure. v3:29 pages, no
figures. v3 includes an account of the relationship between the adjacency
matroid of a graph and the delta-matroid of a graph. v4: 30 pages, 1 figure.
v5: 31 pages, 1 figure. v6: 38 pages, 3 figures. v6 includes a discussion of
the duality between graphic matroids and adjacency matroids of looped circle
graph
Some Applications of the Weighted Combinatorial Laplacian
The weighted combinatorial Laplacian of a graph is a symmetric matrix which is the discrete analogue of the Laplacian operator. In this thesis, we will study a new application of this matrix to matching theory yielding a new characterization of factor-criticality in graphs and matroids. Other applications are from the area of the physical design of very large scale integrated circuits. The placement of the gates includes the minimization of a quadratic form given by a weighted Laplacian. A method based on the dual constrained subgradient method is proposed to solve the simultaneous placement and gate-sizing problem. A crucial step of this method is the projection to the flow space of an associated graph, which can be performed by minimizing a quadratic form given by the unweighted combinatorial Laplacian.Andwendungen der gewichteten kombinatorischen Laplace-Matrix Die gewichtete kombinatorische Laplace-Matrix ist das diskrete Analogon des Laplace-Operators. In dieser Arbeit stellen wir eine neuartige Charakterisierung von Faktor-Kritikalität von Graphen und Matroiden mit Hilfe dieser Matrix vor. Wir untersuchen andere Anwendungen im Bereich des Entwurfs von höchstintegrierten Schaltkreisen. Die Platzierung basiert auf der Minimierung einer quadratischen Form, die durch eine gewichtete kombinatorische Laplace-Matrix gegeben ist. Wir präsentieren einen Algorithmus für das allgemeine simultane Platzierungs- und Gattergrößen-Optimierungsproblem, der auf der dualen Subgradientenmethode basiert. Ein wichtiger Bestandteil dieses Verfahrens ist eine Projektion auf den Flussraum eines assoziierten Graphen, die als die Minimierung einer durch die Laplace-Matrix gegebenen quadratischen Form aufgefasst werden kann