The group of automorphisms of a zero-divisor graph based on rank one upper triangular matrices
The group of automorphisms of a zero-divisor graph based on rank one upper triangular matrices
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DOI：
10.1016/j.laa.2014.07.041
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发表时间：
2014-11
影响因子：
1.1
通讯作者：
Dein Wong;Xiaobin Ma;Jinming Zhou
中科院分区：
文献类型：
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作者：
Dein Wong;Xiaobin Ma;Jinming Zhou
设F q是一个有q个元素的有限域，n（≥2）是一个正整数，Mat n (q)是F q上所有n× n个矩阵的集合，R （n, q）是Mat （n, q）中所有1阶上三角矩阵的集合。matn (q)的零因子图，写为Γ （matn (q)），是一个顶点集matn (q)的所有非零零因子的有向图，并且存在一条从顶点a到顶点B的有向边，写为a→B，当且仅当a B= 0。本文确定了具有顶点集R （n, q）的Γ （matn (q)）的诱导子图Γ （R (n, q)）的自同构。P是可逆上三角矩阵；π是域F q的自同构，[π (x i j)]表示其（i, j）项为π (x i j)的矩阵。
Let F q be a finite field with q elements, n (≥ 2) a positive integer, Mat n (q) the set of all n× n matrices over F q, R (n, q) the set of all rank one upper triangular matrices in Mat (n, q). The zero-divisor graph of Mat n (q), written as Γ (Mat n (q)), is a directed graph with vertex set all nonzero zero-divisors of Mat n (q), and there is a directed edge from a vertex A to a vertex B, written as A→ B, if and only if A B= 0. In this paper, we determine the automorphisms of an induced subgraph, written as Γ (R (n, q)), of Γ (Mat n (q)) with vertex set R (n, q). The main theorem of this article proves that a bijective map σ on R (n, q) with n≥ 3 is an automorphism of Γ (R (n, q)) if and only if σ (X)= a X P− 1 [π (x i j)] P,∀ X=[x i j]∈ R (n, q), where a X∈ F q⁎ depends on X; P is an invertible upper triangular matrix; π is an automorphism of the field F q,[π (x i j)] denotes the matrix whose (i, j)-entry is π (x i j).