Mathematics Faculty Articles

Title

The Minimum Rank of Universal Adjacency Matrices

Document Type

Article

Publication Date

10-15-2012

Publication Title

Linear Algebra and its Applications

Keywords

Adjacency matrix, Universal adjacency matrix, Laplacian matrix, Minimum rank, Graph, Path, Cycle

ISSN

0024-3795

Volume

437

Issue/No.

8

First Page

2064

Last Page

2076

Abstract

In this paper we introduce a new parameter for a graph called the minimum universal rank. This parameter is similar to the minimum rank of a graph. For a graph G the minimum universal rank of G is the minimum rank over all matrices of the form

U(α,β,γ,δ)=αA+βI+γJ+δDU(α,β,γ,δ)=αA+βI+γJ+δD

where A is the adjacency matrix of G, J is the all ones matrix and D is the matrix with the degrees of the vertices in the main diagonal, and α≠0,β,γ,δ are scalars. Bounds for general graphs based on known graph parameters are given, as is a formula for the minimum universal rank for regular graphs based on the multiplicity of the eigenvalues of A. The exact value of the minimum universal rank of some families of graphs are determined, including complete graphs, complete bipartite graph, paths and cycles. Bounds on the minimum universal rank of a graph obtained by deleting a single vertex are established. It is shown that the minimum universal rank is not monotone on induced subgraphs, but bounds based on certain induced subgraphs, including bounds on the union of two graphs, are given.

Comments

Under an Elsevier user license

AMS classification

  • 05C50;
  • 15A03;
  • 15A18;
  • 15A27

DOI

10.1016/j.laa.2012.05.033