Council may order the owner of any property abutting on any public street or alley within this Town to grade that part of the sidewalk of such street abutting on such property, if not then already graded, and to construct thereon a sidewalk pavement, the same to be constructed according to the specifications therefor. Such grade shall conform to the adopted grade of such street. Such order shall designate the width of such sidewalk pavement so ordered to be constructed. The Mayor shall cause notice of such order to be served upon the owner of such property, such notice to be served in the manner provided by the laws of the State for the service of notices generally, or by publication in a newspaper of general circulation published in the County in the manner provided by the laws of the State for the publication of notices generally.