An incidence matrix is a mathematical representation used in graph theory to describe the relationship between vertices and edges in a graph. In this matrix, rows typically represent the vertices, while columns represent the edges. Each entry in the matrix indicates whether a vertex is connected to an edge, often using binary values: 1 for connection and 0 for no connection. This tool is particularly useful for analyzing the structure of graphs, such as those found in networks, social media, or transportation systems. By using an incidence matrix, researchers can easily visualize and manipulate the connections within a graph, aiding in various applications like optimization and network design.