Combinatorics and Graph Theory

2103 Submissions

[2] viXra:2103.0099 [pdf] submitted on 2021-03-16 11:07:48

Chordal Bipartite Graphs Are Rank Determined

Authors: Yaroslav Shitov
Comments: 16 Pages.

A partial matrix A is a rectangular array with entries in F ∪ {∗}, where F is the ground field, and ∗ is a placeholder symbol for entries which are not specified. The minimum rank mr(A) is the smallest value of the ranks of all matrices obtained from A by replacing the ∗ symbols with arbitrary elements in F. For any bipartite graph G with vertices (U, V), one defines the set M(G) of partial matrices in which the row indexes are in U, the column indexes are in V, and the (u, v) entry is specified if and only if u, v are adjacent in G. We prove that, if G is chordal bipartite, then the minimum rank of any matrix in M(G) is determined by the ranks of its fully specified submatrices. This result was conjectured by Cohen, Johnson, Rodman, Woerdeman in 1989.
Category: Combinatorics and Graph Theory

[1] viXra:2103.0053 [pdf] submitted on 2021-03-10 08:33:08

The Theory of Cells

Authors: Theophilus Agama
Comments: 10 Pages.

In this paper we introduce and develop the notion of universe, induced communities and cells with their corresponding spots. We study the concept of the density, the mass of communities, the concentration of spots in a typical cell, connectedness and the rotation of communities. In any case we establish the connection that exist among these notions. We also formulate the celebrated union-close set conjecture in the language of density of spots and the mass of a typical community.
Category: Combinatorics and Graph Theory