# Undergraduate Research Programs in Math

“Coloring graphs, G3/n, with fractional powers” . Willems, E.; Moore, E. School of Mathematics and Sciences, Wayland Baptist University, Plainview, TX, USA.

Mathematicians studying Graph Theory have invested a great deal of time in exploring graph colorings. Recently (2010), Moharram Iradmusa investigated and published a study concerning the chromatic number of fractional power graphs. His paper consists of a few valuable theorems and lemmas which find the chromatic number of specific fractional power graphs. Iradmusa concluded his investigation by stating a conjecture regarding the chromatic number of fractional power graphs.

Conjecture A(m): Let G be a connected graph with ?(G) ≥ 3 and m be a positive integer greater than 1. Then for any positive integer     n ≥ m, χ(G^(m/n) ) = ω(G^(m/n) ) (, p. 1556). Iradmusa proved some special cases of this conjecture. This paper intends to extend progress towards proving the conjecture by demonstrating that χ(G^(3/n) ) = ω(G^(3/n) ) when n is a natural number greater than 3. The proof to this claim requires four cases, n = 4, 5, 6, and 7, and is finished by repetitively utilizing a lemma provided by Iradmusa to inductively demonstrate that χ(G^(3/n) ) = ω(G^(3/n) ) for n ≥ 4.

"Knot selection in least squares approximate with free knot splines." Adamson,B.; Franklin, S. School of Mathematics and Sciences, Wayland Baptist University, Plainview, TX, USA.

The interpolation of data can provide valuable insights into underlying patterns in data, however common methods of interpolation often result in poor curves with sufficiently large or noisy data sets are used; in these cases the interpolating polynomial becomes overly complicated with unrealistic oscillations. The use of least squares approximation with piecewise polynomials and spline's can alleviate some of the complexity, however the selection of the knots or breakpoints is critical to the quality of the approximating curve. Free knot spline's were implemented in order to capture the underlying pattern while seeking to acquire small least square residuals. Several algorithms were developed which utilized various non-linear optimization methods in order to determine the optimal knot sequence. Monte Carlo experiments were used to determine the most effective algorithm for various data sets. We found that the effectiveness of a given algorithm was dependent on the structure of the underlying trends.