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arXiv · 1903.07355

Maximal graphs with respect to rank

H. Esmailian·E. Ghorbani·S. Hossein Ghorban·G. B. Khosrovshahi
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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).

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H. Esmailian, E. Ghorbani, S. Hossein Ghorban, G. B. Khosrovshahi. 2019-03-18. Maximal graphs with respect to rank. https://arxiv.org/abs/1903.07355

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