An extension of Hoffman and Smith's subdivision theorem

TL;DR

This paper extends Hoffman and Smith's subdivision theorem by showing that subdividing high-degree vertices in graphs also strictly decreases the largest eigenvalue, broadening the theorem's applicability.

Contribution

It generalizes the original theorem to include graphs with vertices of degree four or more, not just those with internal paths.

Findings

Largest eigenvalue decreases with subdivision of high-degree vertices

Extension applies to a broader class of graphs

Provides new insights into spectral graph modifications

Abstract

In 1975 Hoffman and Smith showed that for a graph $G\ne\tilde{D}_n$ 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 $G\ne K_{1,4}$ 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.