Orthogonal Representations Of Graphs Research Articles

Positive matching decompositions of graphs

A matching M in a graph Γ is positive if Γ has a vertex-labeling such that M coincides with the set of edges with positive weights. A positive matching decomposition (pmd) of Γ is an edge-partition M 1 , … , M p of Γ such that M i is a positive matching in Γ − ( M 1 ∪ ⋯ ∪ M i − 1 ) , for i = 1 , … , p . The pmds of graphs are used to study algebraic properties of the Lovász–Saks–Schrijver ideals arising from orthogonal representations of graphs. We give a characterization of pmds of graphs in terms of alternating closed walks and apply it to study pmds of various classes of graphs including complete multipartite graphs, (regular) bipartite graphs, cacti, generalized Petersen graphs, etc. We further show that computation of pmds of a graph can be reduced to that of its maximum pendant-free subgraph.

Faithful orthogonal representations of graphs from partition logics

Partition logics often allow a dual probabilistic interpretation: a classical one for which probabilities lie on the convex hull of the dispersion-free weights and another one, suggested independently from the quantum Born rule, in which probabilities are formed by the (absolute) square of the inner product of state vectors with the faithful orthogonal representations of the respective graph. Two immediate consequences are the demonstration that the logico-empirical structure of observables does not determine the type of probabilities alone and that complementarity does not imply contextuality.

On orthogonal representations of graphs

Let G be a simple graph with vertices 1, 2, …, n and F be a field. We consider representation of G by vectors x 1, x 2, …, x n∈F d such that for i≠j the product x ix j=x i 1x j 1+x i 2x j 2+⋯+x i dx j d is equal to zero if and only if vertices i and j are not adjacent in G. The least dimension d necessary for such representations is studied as a function of G.

Sparsity Patterns with High Rank Extremal Positive Semidefinite Matrices

This article concerns the positive semidefinite matrices $M_+ ( G )$ with zero entries in prescribed locations; that is, matrices with given sparsity graph G. The issue here is the rank of the extremals of the cone $M_+ ( G )$. It was shown in [J. Agler, J. W. Helton, S. McCullough, and L. Rodman, Linear Algebra Appl., 107 (1988), pp. 101–149] that the key in constructing high rank extreme points resides in certain atomic graphs G called blocks and superblocks. The k-superblocks are defined to be sparsity graphs G that contain an extreme point of rank k while containing (in an extremely strong sense) no graph with the same property. The goal of this article is to write down all graphs that are superblocks. The article succeeds completely for $k \leq 4$ and it lists necessary conditions in general as well as sufficient conditions. The subject is closely related to orthogonal representations of graphs as studied earlier in [L. Lovász, M. Saks, and A. Schrijver, Linear Algebra Appl., 114/115 (1989), pp. 439–454] and in the previously mentioned paper by Alger et al. Indeed, the paper is an extension of the findings of Alger et al.