In this paper, we explore the connection between sensor networks and graph theory. Sensor networks represent distributed systems of interconnected devices that collect and transmit data, while graph theory provides a robust framework for modeling and analyzing complex networks. Specifically, we focus on vertex coloring, Eulerian paths, and Hamiltonian paths within the Delaunay graph associated with a sensor network. These concepts have critical applications in sensor networks, including connectivity analysis, efficient data collection, route optimization, task scheduling, and resource management. We derive theoretical results related to the chromatic number and the existence of Eulerian and Hamiltonian trails in the graph linked to the sensor network. Additionally, we complement this theoretical study with the implementation of several algorithmic procedures. A case study involving the monitoring of a sugarcane field, coupled with a computational analysis, demonstrates the performance and practical applicability of these algorithms in real-world scenarios.