Near Threshold Graphs

Abstract

A conjecture of Grone and Merris states that for any graph G, its Laplacian spectrum, A(G), is majorized by its conjugate degree sequence, D* (G). That conjecture prompts an investigation of the relationship between A(G) and D* (G), and Merris has characterized the graphs G for which the multisets A(G) and D* (G) are equal. In this paper, we provide a constructive characterization of the graphs G for which A(G) and D* (G) share all but two elements.