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The rank of a graph is defined to be the rank of its adjacency matrix. Agraph is called reduced if it has no isolated vertices and no two vertices withthe same set of neighbors. A reduced graph $G$ is said to be maximal if anyreduced graph containing $G$ as a proper induced subgraph has a higher rank.The main intent of this paper is to present some results on maximal graphs.First, we introduce a characterization of maximal trees (a reduced tree is amaximal tree if it is not a proper subtree of a reduced tree with the samerank). Next, we give a near-complete characterization of maximal `generalizedfriendship graphs.' Finally, we present an enumeration of all maximal graphswith ranks $8$ and $9$. The ranks up to $7$ were already done by Lepovi\'c(1990), Ellingham (1993), and Lazi\'c (2010).

Maximal graphs with respect to rank