Related papers: Perfect Elimination Orderings for Symmetric Matric…
An important property of chordal graphs is that these graphs are characterized by existence of perfect elimination orderings on their vertex sets. In this paper, we generalize the notion of perfect elimination orderings to signed graphs,…
We study deterministic constructions of graphs for which the unique completion of low rank matrices is generically possible regardless of the values of the entries. We relate the completability to the presence of some patterns (particular…
This work introduces a new class of symmetric matrix structures, called harmonic structures, which enable the generation of all possible directed transitions $(x_i, x_{i+1})$ over a set of $n$ symbols, without internal repetitions. Unlike…
We define strongly chordal digraphs, which generalize strongly chordal graphs and chordal bipartite graphs, and are included in the class of chordal digraphs. They correspond to square 0,1 matrices that admit a simultaneous row and column…
We study a class of parametrizations of convex cones of positive semidefinite matrices with prescribed zeros. Each such cone corresponds to a graph whose non-edges determine the prescribed zeros. Each parametrization in this class is a…
We introduce a remarkable new family of norms on the space of $n \times n$ complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their construction and validation involve probability theory,…
We survey the theory of totally symmetric sets, with applications to homomorphisms of symmetric groups, braid groups, linear groups, and mapping class groups.
We study the class of 1-perfectly orientable graphs, that is, graphs having an orientation in which every out-neighborhood induces a tournament. 1-perfectly orientable graphs form a common generalization of chordal graphs and circular arc…
Matrix partition problems generalize a number of natural graph partition problems, and have been studied for several standard graph classes. We prove that each matrix partition problem has only finitely many minimal obstructions for split…
Symmetry breaking for graphs and other combinatorial objects is notoriously hard. On the one hand, complete symmetry breaks are exponential in size. On the other hand, current, state-of-the-art, partial symmetry breaks are often considered…
A totally symmetric set is a finite subset of a group for which any permutation of the elements can be realized by conjugation in the ambient group. Such sets are rigid under homomorphisms, and so exert a great deal of control over the…
We consider constrained variants of graph homomorphisms such as embeddings, monomorphisms, full homomorphisms, surjective homomorpshims, and locally constrained homomorphisms. We also introduce a new variation on this theme which derives…
We study the question of whether, for a given class of finite graphs, one can define, for each graph of the class, a linear ordering in monadic second-order logic, possibly with the help of monadic parameters. We consider two variants of…
A symmetric matrix is Robinsonian if its rows and columns can be simultaneously reordered in such a way that entries are monotone nondecreasing in rows and columns when moving toward the diagonal. The adjacency matrix of a graph is…
Vine copulas can efficiently model multivariate probability distributions. This paper focuses on a more thorough understanding of their structures, since in the literature, vine copula representations are often ambiguous. The graph…
Recently, there has been interest in the question of whether a partial matrix in which many of the fully defined principal submatrices are PSD is approximately PSD completable. These questions are related to graph theory because we can…
A matroid is supersolvable if it has a maximal chain of flats each of which is modular. A matroid is saturated if every round flat is modular. In this article we present supersolvable saturated matroids as analogues to chordal graphs, and…
This paper is concerned with structures of general graphs with perfect matchings. We first reveal a partially ordered structure among factor-components of general graphs with perfect matchings. Our second result is a generalization of…
Cylindrical algebraic decomposition (CAD) plays an important role in the field of real algebraic geometry and many other areas. As is well-known, the choice of variable ordering while computing CAD has a great effect on the time and memory…
In this paper, we define and study absolute matrix order ideals in absolute matrix order unit spaces. As an application of absolute matrix order unit property, we construct some kinds of absolute matrix order ideals in absolute matrix order…