What do the house, Miss Emily, and the South have in common with one another?
Miss Emily, her home, and the South all share some similarities, including their resistance to change and their eventual erosion.
Miss Emily does not like change and resists it greatly! She still believes Colonel Sartoris is still alive, for example, but he's been dead for 10 years (she refuses to pay her taxes). Also, Miss Emily makes her home resistant to change, as well. She leaves it as is and it is coated in dust and is an eyesore in town due to the fact she has left it unchanged and doesn't do improvements on it, etc. Finally, this could also relate to the "Old South," which was slowly changing and becoming more and more of an industrialized area instead of an agricultural area, which many people fought very hard against.
Also, all 3 erode and deteriorate. The Old South suffered times of extreme poverty as it switched from agriculture to industry. Miss Emily's appearance suffered, as well, as she turned from a beautiful young woman into a portly, pale old woman. The house also deteriorates and becomes an eyesore in the town.