#### Campus Units

Mathematics

#### Document Type

Article

#### Publication Version

Published Version

#### Publication Date

12-2014

#### Journal or Book Title

Electronic Journal of Linear Algebra

#### Volume

27

#### First Page

907

#### Last Page

934

#### DOI

10.13001/1081-3810.2007

#### Abstract

A loop graph S is a finite undirected graph that allows loops but does not allow multiple edges. The set S(S) of real symmetric matrices associated with a loop graph of order n is the set of symmetric matrices A = [a(ij)] is an element of R-nxn such that a(ij) not equal 0 if and only if ij is an element of E(S). The minimum (maximum) rank of a loop graph is the minimum (maximum) of t he ranks of the matrices in S(S). Loop graphs having minimum rank at most two are characterized (by forbidden induced subgraphs and graph complements) and loop graphs having minimum rank equal to the order of the graph are characterized. A Schur complement reduction technique is used to determine the minimum ranks of cycles with various loop configurations; the minimum ranks of complete graphs and paths with various configurations of loops are also determined. Unlike simple graphs, loop graphs can have maximum rank less than the order of the graph. Some results are presented on maximum rank and which ranks between minimum and maximum can be realized. Interesting open questions remain.

#### Copyright Owner

The Author(s)

#### Copyright Date

2014

#### Language

en

#### File Format

application/pdf

#### Recommended Citation

Bozeman, Chassidy; Ellsworth, AnnaVictoria; Hogben, Leslie; Lin, Jephian Chin-Hung; Maurer, Gabi; Nowak, Kathleen; Rodriguez, Aaron; and Strickland, James, "Minimum Rank of Graphs with Loops" (2014). *Mathematics Publications*. 231.

https://lib.dr.iastate.edu/math_pubs/231

## Comments

This article is published as Bozeman, Chassidy, AnnaVictoria Ellsworth, Leslie Hogben, Jephian Chin-Hung Lin, Gabi Maurer, Kathleen Nowak, Aaron Rodriguez, and James Strickland. "Minimum rank of graphs with loops."

The Electronic Journal of Linear Algebra27 (2014): 907-934. DOI: 10.13001/1081-3810.2007. Posted with permission.