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Register a new userOriented Hypergraphic Matrix-tree Type Theorems and Bidirected Minors via Boolean Order Ideals
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Restrictions of incidence-preserving path maps produce an oriented hypergraphic All Minors Matrix-tree Theorems for Laplacian and adjacency matrices. The images of these maps produce a locally signed graphic, incidence generalization, of cycle covers and basic figures that correspond to incidence-k-forests. When restricted to bidirected graphs the natural partial ordering of maps results in disjoint signed boolean lattices whose minor calculations correspond to principal order ideals. As an application, (1) the determinant formula of a signed graphic Laplacian is reclaimed and shown to be determined by the maximal positive-circle-free elements, and (2) spanning trees are equivalent to single-element order ideals.
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An oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure. In this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic Laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs. Additionally, we provide bounds on the determinant and permanent of the Laplacian matrix, characterize the oriented hypergraphs in which the upper bound is sharp, and demonstrate that the lower bound is never achieved.
An oriented hypergraph is an oriented incidence structure that allows for the generalization of graph theoretic concepts to integer matrices through its locally signed graphic substructure. The locally graphic behaviors are formalized in the subobject classifier of incidence hypergraphs. Moreover, the injective envelope is calculated and shown to contain the class of uniform hypergraphs -- providing a combinatorial framework for the entries of incidence matrices. A multivariable all-minors characteristic polynomial is obtained for both the determinant and permanent of the oriented hypergraphic Laplacian and adjacency matrices arising from any integer incidence matrix. The coefficients of each polynomial are shown to be submonic maps from the same family into the injective envelope limited by the subobject classifier. These results provide a unifying theorem for oriented hypergraphic matrix-tree-type and Sachs-coefficient-type theorems. Finally, by specializing to bidirected graphs, the trivial subclasses for the degree-$k$ monomials of the Laplacian are shown to be in one-to-one correspondence with $k$-arborescences.
An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. We define the adjacency, incidence and Laplacian matrices of an oriented hypergraph and study each of them. We extend several matrix results known for graphs and signed graphs to oriented hypergraphs. New matrix results that are not direct generalizations are also presented. Finally, we study a new family of matrices that contains walk information.
Tree-width and its linear variant path-width play a central role for the graph minor relation. In particular, Robertson and Seymour (1983) proved that for every tree~$T$, the class of graphs that do not contain $T$ as a minor has bounded path-width. For the pivot-minor relation, rank-width and linear rank-width take over the role from tree-width and path-width. As such, it is natural to examine if for every tree~$T$, the class of graphs that do not contain $T$ as a pivot-minor has bounded linear rank-width. We first prove that this statement is false whenever $T$ is a tree that is not a caterpillar. We conjecture that the statement is true if $T$ is a caterpillar. We are also able to give partial confirmation of this conjecture by proving: (1) for every tree $T$, the class of $T$-pivot-minor-free distance-hereditary graphs has bounded linear rank-width if and only if $T$ is a caterpillar; (2) for every caterpillar $T$ on at most four vertices, the class of $T$-pivot-minor-free graphs has bounded linear rank-width. To prove our second result, we only need to consider $T=P_4$ and $T=K_{1,3}$, but we follow a general strategy: first we show that the class of $T$-pivot-minor-free graphs is contained in some class of $(H_1,H_2)$-free graphs, which we then show to have bounded linear rank-width. In particular, we prove that the class of $(K_3,S_{1,2,2})$-free graphs has bounded linear rank-width, which strengthens a known result that this graph class has bounded rank-width.
In an earlier paper, the first two authors defined orientations on hypergraphs. Using this definition we provide an explicit bijection between acyclic orientations in hypergraphs and faces of hypergraphic polytopes. This allows us to obtain a geometric interpretation of the coefficients of the antipode map in a Hopf algebra of hypergraphs. This interpretation differs from similar ones for a different Hopf structure on hypergraphs provided recently by Aguiar and Ardila. Furthermore, making use of the tools and definitions developed here regarding orientations of hypergraphs we provide a characterization of hypergraphs giving rise to simple hypergraphic polytopes in terms of acyclic orientations of the hypergraph. In particular, we recover this fact for the nestohedra and the hyper-permutahedra, and prove it for generalized Pitman-Stanley polytopes as defined here.
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