Degree maximal graphs are Laplacian integral

Abstract

An integer sequence (d) = (d1, d2,…,dn) is graphic if there is a graph whose degree sequence is (d). A graph G is maximal if its degree sequence is majorized by no other graphic sequence. The Laplacian matrix of G is L(G) = D(G) - A(G), where D(G) is the diagonal matrix of vertex degrees and A(G) is the (0,1) adjacency matrix. The article contains an explicit (algorithmic) construction of all maximal graphs, from which it follows, e.g., that, apart from 0, the Laplacian spectrum of a maximal graph is the conjugate of its degree sequence.