Maximal graphs with respect to rank

TL;DR

This paper investigates the structure of maximal graphs with respect to their adjacency matrix rank, providing characterizations for trees and generalized friendship graphs, and enumerating maximal graphs of ranks 8 and 9.

Contribution

It offers new characterizations of maximal trees and generalized friendship graphs, and enumerates maximal graphs for ranks 8 and 9, extending previous classifications.

Findings

Characterization of maximal trees as non-extendable reduced trees with the same rank.

Near-complete classification of maximal generalized friendship graphs.

Enumeration of all maximal graphs with ranks 8 and 9.

Abstract

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. A reduced graph $G$ is said to be maximal if any reduced 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 a maximal tree if it is not a proper subtree of a reduced tree with the same rank). Next, we give a near-complete characterization of maximal `generalized friendship graphs.' Finally, we present an enumeration of all maximal graphs with ranks $8$ and $9$. The ranks up to $7$ were already done by Lepovi\'c (1990), Ellingham (1993), and Lazi\'c (2010).