## Minimum Rank of Graphs with Loops

2014-12-01
##### Authors
Bozeman, Chassidy
Ellsworth, AnnaVictoria
Lin, Jephian
Maurer, Gabi
Nowak, Kathleen
Rodriguez, Aaron
Strickland, James
Person
Hogben, Leslie
Associate Dean
##### Organizational Units
Organizational Unit
Electrical and Computer Engineering

The Department of Electrical and Computer Engineering (ECpE) contains two focuses. The focus on Electrical Engineering teaches students in the fields of control systems, electromagnetics and non-destructive evaluation, microelectronics, electric power & energy systems, and the like. The Computer Engineering focus teaches in the fields of software systems, embedded systems, networking, information security, computer architecture, etc.

History
The Department of Electrical Engineering was formed in 1909 from the division of the Department of Physics and Electrical Engineering. In 1985 its name changed to Department of Electrical Engineering and Computer Engineering. In 1995 it became the Department of Electrical and Computer Engineering.

Dates of Existence
1909-present

Historical Names

• Department of Electrical Engineering (1909-1985)
• Department of Electrical Engineering and Computer Engineering (1985-1995)

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Organizational Unit
Mathematics
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##### 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.