Maximal graphs and graphs with maximal spectral radius Original Research Article

A maximal graph is a connected graph with degree sequence not majorized by the degree sequence of any other graph. Relationships between three different sequences of integers that describe maximal graphs are given. Recursive and explicit forms of the characteristic polynomial of the adjacency matrix of a maximal graph are derived. A correspondence between maximal graphs and connected graphs with stepwise adjacency matrices shows that, among all connected graphs with n vertices and e edges, the graph with maximal spectral radius is a maximal graph. Such maximal graphs are identified for certain values of n and e.