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.