Degree Type

Dissertation

Date of Award

2018

Degree Name

Doctor of Philosophy

Department

Mathematics

Major

Applied Mathematics

First Advisor

Michael Young

Second Advisor

Leslie Hogben

Abstract

Exponential domination in graphs evaluates the influence that a particular vertex exerts on the remaining vertices within a graph. The amount of influence a vertex exerts is measured through an exponential decay formula with a growth factor of one-half. An exponential dominating set consists of vertices who have significant influence on other vertices. In non-porous exponential domination, vertices in an exponential domination set block the influence of each other. Whereas in porous exponential domination, the influence of exponential dominating vertices are not blocked. For a graph $G,$ the non-porous and porous exponential domination numbers, denoted $\gamma_e(G)$ and $\gamma_e^*(G),$ are defined to be the cardinality of the minimum non-porous exponential dominating set and cardinality of the minimum porous exponential dominating set, respectively. This dissertation focuses on determining lower and upper bounds of the non-porous and porous exponential domination number of the King grid $\mathcal{K}_n,$ Slant grid $\mathcal{S}_n,$ $n$-dimensional hypercube $Q_n,$ and the general consecutive circulant graph $C_{n,[\ell]}.$

A method to determine the lower bound of the non-porous exponential domination number for any graph is derived. In particular, a lower bound for $\gamma_e^*(Q_n)$ is found. An upper bound for $\gamma_e^*(Q_n)$ is established through exploiting distance properties of $Q_n.$ For any grid graph $G,$ linear programming can be incorporated with the lower bound method to determine a general lower bound for $\gamma_e^*(G).$ Applying this technique to the grid graphs $\mathcal{K}_n$ and $\mathcal{S}_n$ yields lower bounds for $\gamma_e^*(\mathcal{K}_n)$ and $\gamma_e^*(\mathcal{S}_n).$ Upper bound constructions for $\gamma_e^*(\mathcal{K}_n)$ and $\gamma_e^*(\mathcal{S}_n)$ are also derived. Finally, it is shown that $\gamma_e(C_{n,[\ell]}) = \gamma_e^*(C_{n,[\ell]}).$

DOI

https://doi.org/10.31274/etd-180810-5968

Copyright Owner

Michael Dairyko

Language

en

File Format

application/pdf

File Size

79 pages

Included in

Mathematics Commons

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