The regular graph of the space of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n \times m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation> matrices over a field <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">F</mi> </math></EquationSource> </InlineEquation> is defined as the undirected graph whose vertices are matrices of rank <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\min (n, m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">min</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and distinct matrices A and B are connected by an edge if and only if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\text {rk}(A + B) < \min (n,m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>rk</mtext> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo><</mo> <mo movablelimits="true">min</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$|\mathbb {F}| > 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="double-struck">F</mi> <mo stretchy="false">|</mo> <mo>></mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$m, n \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, all additive automorphisms of the regular graphs are characterized. Furthermore, it is proved that any automorphism of the regular graph preserves the rank distance <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$d(A, B) = \text {rk}(A - B)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mtext>rk</mtext> <mo stretchy="false">(</mo> <mi>A</mi> <mo>-</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Bibliography: 11 titles.

ADDITIVE AUTOMORPHISMS OF REGULAR MATRIX GRAPH

期刊论文
发表日期：

2026年5月19日

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I. I. Gusev,
A. M. Maksaev,
V. V. Promyslov

摘要

The regular graph of the space of $n \times m$ $n \times m$ matrices over a field $\mathbb {F}$ $F$ is defined as the undirected graph whose vertices are matrices of rank $\min (n, m)$ $min (n, m)$ , and distinct matrices A and B are connected by an edge if and only if $\text {rk}(A + B) < \min (n,m)$ $rk (A + B) < min (n, m)$ . In this paper, for $|\mathbb {F}| > 4$ $| F | > 4$ and $m, n \ge 2$ $m, n \geq 2$ , all additive automorphisms of the regular graphs are characterized. Furthermore, it is proved that any automorphism of the regular graph preserves the rank distance $d(A, B) = \text {rk}(A - B)$ $d (A, B) = rk (A - B)$ . Bibliography: 11 titles.