6 research outputs found
Graphs with few trivial characteristic ideals
We give a characterization of the graphs with at most three trivial characteristic ideals. This implies the complete characterization of the regular graphs whose critical groups have at most three invariant factors equal to 1 and the characterization of the graphs whose Smith groups have at most 3 invariant factors equal to 1. We also give an alternative and simpler way to obtain the characterization of the graphs whose Smith groups have at most 3 invariant factors equal to 1, and a list of minimal forbidden graphs for the family of graphs with Smith group having at most 4 invariant factors equal to 1
Constructions of cospectral graphs with different zero forcing numbers
Several researchers have recently explored various graph parameters that can
or cannot be characterized by the spectrum of a matrix associated with a graph.
In this paper we show that several NP-hard zero forcing numbers are not
characterized by the spectra of several types of associated matrices with a
graph. In particular, we consider standard zero forcing, positive semidefinite
zero forcing, and skew zero forcing, and provide constructions of infinite
families of pairs of cospectral graphs which have different values for these
numbers. We explore several methods for obtaining these cospectral graphs
including using graph products, graph joins, and graph switching. Among these,
we provide a construction involving regular adjacency cospectral graphs; the
regularity of this construction also implies cospectrality with respect to
several other matrices including the Laplacian, signless Laplacian, and
normalized Laplacian. We also provide a construction where pairs of cospectral
graphs can have an arbitrarily large difference between their zero forcing
numbers
The Strong Spectral Property of Graphs: Graph Operations and Barbell Partitions
The utility of a matrix satisfying the Strong Spectral Property has been well
established particularly in connection with the inverse eigenvalue problem for
graphs. More recently the class of graphs in which all associated symmetric
matrices possess the Strong Spectral Property (denoted ) were studied,
and along these lines we aim to study properties of graphs that exhibit a
so-called barbell partition. Such a partition is a known impediment to
membership in the class . In particular we consider the existence of
barbell partitions under various standard and useful graph operations
Arithmetical structures on dominated polynomials
In~\cite{algorithmic} was given an algorithm that computes arithmetical
structures on matrices. We use some of the ideas contained there to get an
algorithm that computes arithmetical structures over dominated polynomials. A
dominated polynomial is an integer multivariate polynomial such that contains a
monomial which is divided by all its monomials.Comment: This article correspond to an unpublished part of article
arXiv:2101.0523