Additive automorphisms of regular matrix graph

I. I. Gusev^a, A. M. Maksaev^ab, V. V. Promyslov<sup>a

a</sup> National Research University Higher School of Economics, Moscow
^b Moscow Center for Fundamental and Applied Mathematics

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

Key words and phrases: rectangular matrices, graph automorphism, regular graph.

UDC: 512.643

Received: 10.10.2025