The tomato sandwich is the extreme case of how little a sandwich can be, and naming the mayonnaise is the South's way of saying the little that is there has to be exactly right. A thick slice of a ripe summer tomato, salt, mayonnaise, and soft white bread. That is the whole sandwich, and with nothing to hide behind, the mayonnaise stops being a background condiment and becomes a load-bearing flavor. Duke's is specified because it is built tangy and unsweetened, with a high egg-yolk richness and no sugar, so it sharpens the tomato instead of rounding it off the way a sweeter spread would. The choice is not brand loyalty for its own sake; it is a judgment about acid against fruit.
The craft is in restraint and timing. The tomato must be dead ripe and at room temperature, sliced thick enough to be the substance of the sandwich and salted directly so the salt draws and seasons its juice. The mayonnaise is spread edge to edge on both slices, where it does double work: it carries the tang that lifts the tomato, and it waterproofs the crumb just enough to keep the bread from dissolving under a bleeding slice. The bread is deliberately soft, plain, and a little sweet, because a crust with real chew would fight a filling that has none and a sturdier loaf would overwhelm the fruit. There is no toasting and no delay; this is made and eaten in the same few minutes, standing over a sink in tomato season, because the moment the bread starts to go the sandwich is over.
There is not much to vary without becoming a different sandwich, which is the point of it. A few cracks of black pepper or a swap to a softer regional white bread stay inside the form; adding bacon turns it toward a BLT and acquires the structure problems the BLT has; the broader lunch-counter shelf of bound-salad and simple sliced-bread sandwiches runs alongside it. Those are their own builds and deserve their own articles rather than being crowded in here.