The vertical intercept of a weight-mass graph represents the weight of an object when its mass is zero, which is zero weight. This indicates that weight is directly proportional to mass, as weight is the force exerted by gravity on an object. On Earth, an object cannot have weight without mass, reinforcing the concept that both are intrinsically linked. The discussion emphasizes that the y-intercept being zero aligns with the fundamental relationship between mass and weight. Understanding this relationship is crucial for interpreting weight-mass graphs accurately.