An extension of Hoffman and Smith's subdivision theorem

In 1975 Hoffman and Smith showed that for a graph $G\mathrm{\ne }{\tilde{D}}_n$ with an internal path, the value of the largest eigenvalue decreases strictly each time we subdivide the internal path. In this paper we extend this result to show that for a graph $G\mathrm{\ne }{K}_{1,4}$ with a vertex of degree 4 or more, we can subdivide said vertex to create an internal path and the value of the largest eigenvalue also strictly decreases.