#### Degree Name

MS (Master of Science)

#### Program

Mathematical Sciences

#### Date of Award

8-2012

#### Committee Chair or Co-Chairs

Debra J. Knisley

#### Committee Members

Robert A. Beeler, Arial Cintron-Aries

#### Abstract

A graph *G* is a (*t*, *r*)-regular graph if every collection of *t* independent vertices is collectively adjacent to exactly *r* vertices. If a graph *G* is (2, *r*)-regular where *p*, *s*, and *m* are positive integers, and *m* ≥ 2, then when *n* is sufficiently large, then *G* is isomorphic to *G = K _{s}+mK_{p}*, where 2(

*p*-1)+

*s*=

*r*. A nested (2,

*r*)-regular graph is constructed by replacing selected cliques with a (2,

*r*)-regular graph and joining the vertices of the peripheral cliques. For example, in a nested '

*s*' graph when

*n = s + mp*, we obtain

*n = s*. The nested '

_{1}+m_{1}p_{1}+mp*s*' graph is now of the form

*G*. We examine the network properties such as the average path length, clustering coefficient, and the spectrum of these nested graphs.

_{s}= K_{s1}+m_{1}K_{p1}+mK_{p}#### Document Type

Thesis - Open Access

#### Recommended Citation

Brooks, Josh Daniel, "Nested (2,r)-regular graphs and their network properties." (2012). *Electronic Theses and Dissertations.* Paper 1471. http://dc.etsu.edu/etd/1471

#### Copyright

Copyright by the authors.