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<p>The total graph of the space of <i>m</i> × <i>n</i> matrices over a field 𝔽 is the graph with the set of vertices <i>M</i><sub><i>m</i>×<i>n</i></sub>(𝔽) in which distinct matrices <i>A</i> and <i>B</i> are connected by an edge if and only if rank(<i>A</i>+<i>B</i>) &lt; min(<i>m</i>, <i>n</i>). It is proved that over a field of order <i>q</i>, where <i>q</i> is a power of an odd prime, the clique number of the total graph of 2 × <i>n</i> matrices is <i>q</i><sup>n</sup>, whereas that of 3 × 3 matrices is <i>O</i>(<i>q</i><sup>6</sup>). Up to now, this issue has only been examined for 2 × 2 matrices.</p>

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Clique Numbers of the Total Graphs of the 2 × n and 3 × 3 Matrices

  • 期刊论文

  • 发表日期：

    2025年3月1日

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  • A. M. Maksaev,
  • V. V. Promyslov,
  • D. S. Sheshenya

摘要

The total graph of the space of m × n matrices over a field 𝔽 is the graph with the set of vertices Mm×n(𝔽) in which distinct matrices A and B are connected by an edge if and only if rank(A+B) < min(m, n). It is proved that over a field of order q, where q is a power of an odd prime, the clique number of the total graph of 2 × n matrices is qn, whereas that of 3 × 3 matrices is O(q6). Up to now, this issue has only been examined for 2 × 2 matrices.