This thesis provides a fresh perspective on the (vertex) integrity of graphs, serving as a measure of network robustness. The study begins by introducing fundamental concepts and methods for evaluating the integrity of different graph families. An Integer Linear Programming (ILP) model, specifically designed for assessing integrity, is then presented. By applying this model, integrity values are calculated for various graph families, and patterns within the results are identified. These patterns contribute to establishing boundaries or determining exact values of integrity for the analyzed graph families. The analyzed graph families encompass Glued Paths, generalized Theta graphs, (Double) Cone graphs, Fan graph, Lollipop graphs, generalized Barbell graphs, (Dutch) Windmill graphs, Paley graphs, and Kneser graphs. Additionally, two conjectures are formulated: one concerning a lower bound for the integrity of all Paley graphs, and another addressing the exact integrity values of specific Kneser graphs. The ILP model proves to be a valuable tool for further exploration of graph family integrity, offering opportunities for additional research and expanding our understanding of network robustness.