Oriented hypergraphic matrix-tree type theorems and bidirected minors via Boolean order ideals

Abstract

Restrictions of incidence preserving path maps produce 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.

Characterizing attitudinal network graphs through frustration cloud

“…The upper limit on the number of spanning trees is computed by Cayley's theorem: the complete graph with v vertices has v v−2 spanning trees, and a complete bipartite graph with v, q vertices has v q−1 • q v−1 spanning trees (Buekenhout and Parker 1998;Rusnak et al 2018). For a small social network such as the Highland Tribes (Read 1954) in Fig.…”

Generalizing Kirchhoff laws for Signed Graphs

“…Due to the nature of the incidence-maps it is possible for a path to fold back on itself creating a backstep of the form v, i, e, i, v -these are the entries in the hypergraphic degree matrix. A contributor can be regarded as a permutation clone that is a generalized cycle covers similar to Sachs' Theorem to determine characteristic polynomial coefficients [8,1,7]; contributors naturally form Boolean lattices when G is a bidirected graph [14]. The set of contributors of an oriented hypergraph is denoted C(G).…”

“…Two examples appear in Figure 6. The concept of tail-equivalence is a generalization of circle activation classes of bidirected graphs in [14],…”

“…Let A(G) denote a tail-equivalence class of G. As with restricted and reduced contributors we let A(u; w; G) be the elements of tail-equivalency class A(G) where u i ↦ w i , and Â(u; w; G) be the elements of A(u; w; G) with u i ↦ w i removed for each i. From [14], the activation classes and their restricted subclasses (order ideals) of a bidirected graph are Boolean. It was also shown in [10] that the reduced contributors in single element activation classes Â≠0 (u; w; L 0 (G)) are unpacking equivalent to k-arborescences.…”

“…This is accomplished by using the incidence-theoretic approach introduced in [13] to study hypergraphic Laplacians, and the incidence-path mapping families, called contributors, from [7] that generalize cycle covers to classify various hypergraphic characteristic polynomials similar to Sachs' Theorem [1,8]. It was shown in [14] that if all edges are size 2 these generalized cycle covers form Boolean lattices that generalize the Matrix-tree theorem. These Boolean families are naturally cancellative when G is a graph with the trivial single-element classes corresponding to spanning trees -providing "conservation" for the graphic Kirchhoff Laws.…”