Subdivision and graph eigenvalues
Hitesh Kumar, Bojan Mohar, Shivaramakrishna Pragada, Hanmeng Zhan · Linear Algebra and its Applications · 2025
This paper investigates the asymptotic nature of graph spectra when some edges of a graph are subdivided sufficiently many times. In the special case where all edges of a graph are subdivided, we find the exact limits of the k -th largest and k -th smallest eigenvalues for any fixed k . Given a graph, we show that after subdividing sufficiently many times, all but O ( 1 ) eigenvalues of the new graph will lie in the interval [ − 2 , 2 ] . We examine the eigenvalues of the new graph outside this interval, and we prove several results that might be of independent interest.