Research Overview

This ongoing project is part of my undergraduate research in graph theory, focusing on probabilistic processes on graphs, particularly spreading and coloring behaviors in structured networks. I chose to pursue this project because of my strong interest in discrete mathematics and the power that graph-based models have in representing real-world phenomena such as epidemic spread or information diffusion through a network.

The research was carried out under the guidance of Dr. Soumya Bhoumik and Dr. Paul Flesher, where I worked independently while drawing upon discussions and feedback from these mentors. I am grateful for the support provided by the mathematics faculty and department at Fort Hays State University.

Objectives

The primary objectives of this research were to:  

• Analyze the irreversible k-threshold process on corona product graphs of the form Cn ⊙ Kp , determining the smallest sets that fully color the graph.

• Derive closed-form expressions and recurrence relations for the irreversible k-conversion number Ck (Cn ⊙ Kp), which quantifies the size of a minimal coloring seed set.

• Introduce a probabilistic model to compute the probability that a randomly selected seed set of minimum size results in complete coloring, which has not been explored in this context.

• Define the structure of a new graph product, the double corona product, and analyze this product (of cycles and complete graphs) depicted by Cn ⊚ Kp​ . This extension increases structural complexity while preserving graph invariants such as order and size, since Cn ⊙ Kp​ and Cn ⊚ Kp-1 are equal in both vertex and edge count. 

• Extend the deterministic irreversible k-conversion framework to this new graph family and determine exact formulas for Ck (Cn ⊚ Kp) under each possible threshold condition.

Results

Despite some corona and double corona graphs sharing the same number of vertices and edges, their spreading behavior differed significantly. Exact results were established for all k, n, and p values, and the sensitivity of propagation to small changes in k was notably higher in the double corona.

This project fills a gap in the literature by being the first to thoroughly study corona products under the irreversible k-threshold processes, the first to incorporate probabilistic analysis of seeding strategies in this setting, as well as introducing and analyzing the double corona product.

These findings expand the theory of irreversible processes on product graphs and suggest new directions for studying network resilience and stochastic influence models.

More details, graph examples, and proofs are available in the Posters, Presentations, and Academic Writing sections of the Academic Portfolio tab.

Explore the Corona Product & Double Corona Product Graph Visualizer.

My involvement in this project is supported by the 2025 Research Experience for Undergraduates (REU) at the University of Nebraska-Lincoln; funding for this project was provided in part by the National Science Foundation (Award EEC-2349859, REU Site: Sustainability and Resilience of Civil and Environmental Infrastructure in Rural Areas).

Originally developed through a collaboration between UNL, MITRE, the University of Alabama, and FAMU-FSU College of Engineering, the project aims to leverage roadside camera feeds and deep learning models to monitor trucks in freeway work zones and generate real-time trajectory data that could eventually feed into safety alert systems. The project was supported by the Federal Motor Carrier Safety Administration (FMCSA). 

As part of the 2025 REU team (led by Dr. Li Zhao, Dr. Nathan Huynh, and Jiahe Cao) I am focused on improving the realism of the co-simulation environment introducing speed reductions and more natural driving behaviors to better represent real-world work zone conditions. These enhancements are expected to impact future detection and tracking results and provide more generalizable and field-representative findings.

Objectives

The core objective of the project is to develop a robust system for detecting and tracking trucks in work zones using roadside video, with the following technical goals:

• Develop a co-simulation platform combining CARLA and SUMO to replicate a freeway work zone with realistic vehicle dynamics.

• Implement camera-based vehicle detection and tracking using YOLOv8 and ByteTrack.

• Convert 2D image detections into accurate 3D world trajectories using the pinhole camera model and intrinsic/extrinsic calibration.

• Evaluate detection confidence and tracking accuracy under varying traffic volumes and truck densities.

• Identify performance limits and practical trade-offs in trajectory-based warning systems for work zone safety.

Results

Results show that while the detection system handles cars with high accuracy, it struggles more with trucks, especially in dense traffic. Detection confidence for cars remained consistently high (90.7%), while truck detection was lower (85.2%) and more variable. Tracking results followed a similar pattern: cars had an average deviation of 0.880 meters, compared to 1.578 meters for trucks.

These differences were most pronounced as vehicles neared the camera or during high-volume traffic, where larger vehicles introduced occlusion and distorted bounding boxes. For example, truck tracking deviation increased sharply within 40 meters of the camera, while car tracking stayed relatively stable.

Together, these findings highlight a key tradeoff: the system is highly effective under moderate traffic conditions and for smaller vehicles, but more refinement is needed to handle the complexity of tracking trucks in crowded, real-world work zones.

View the corresponding poster and research paper in the Academic Portfolio tab.           [Available Soon]

This portfolio was completed as part of the GSCI674 Aerial Photography and Remote Sensing course at Fort Hays State University and provides a comprehensive exploration of key concepts and techniques in remote sensing. It combines historical background, theoretical principles, and practical applications through ten structured lab exercises and a final independent project.

Each section is supported by hands-on work in ENVI software, allowing me to process, interpret, and analyze remotely sensed imagery. The portfolio also includes a current event report linking course content to real-world applications.

Objectives

The primary objective of this portfolio was to build technical skills and conceptual understanding through structured labs and applied analysis. Each lab focused on a specific area of remote sensing:

• Lab 1: Introduced the ENVI interface and basic tools

• Lab 2: Explored electromagnetic spectra and Landsat band combinations

• Lab 3: Applied atmospheric corrections, orthorectification, and image registration

• Lab 4: Practiced enhancement methods such as histogram adjustments and spectral sharpening

• Lab 5: Introduced supervised and unsupervised image classification

• Lab 6: Used change detection and time series analysis to observe landscape dynamics

• Lab 7: Focused on elevation modeling and LiDAR interpretation

• Lab 8: Covered image fusion techniques for integrating multiple data types

• Lab 9: Applied vegetation indices and water depth estimation methods

• Lab 10: Introduced hyperspectral data processing and analysis

These exercises progressively developed my ability to interpret geospatial data, perform meaningful analyses, and apply remote sensing tools to environmental and land use studies.

Results

By the end of the course, I gained a strong foundation in both the science and application of remote sensing. I became confident in using ENVI for image classification, change detection, and data visualization. Each lab built on the last, leading to a deeper understanding of how remote sensing supports environmental monitoring, land use planning, and resource management.

The final project Analyzing Wetland and Vegetation Change in the Nebraska Sandhills, demonstrates my ability to carry out an independent analysis using the techniques learned. Overall, this portfolio reflects significant growth in technical skills, critical thinking, and the practical use of remote sensing tools for real-world challenges.

View the final guide in the Academic Writing section of the Academic Portfolio tab.