Journal of Algebra and Its Applications ( IF 0.5 ) Pub Date : 2021-07-15 , DOI: 10.1142/s0219498822501936 Avinash Patil 1 , Kiran Shinde 2
The zero-divisor graph of a commutative ring is the graph whose vertices are the nonzero zero divisors in and two vertices and are adjacent if and only if . We study the adjacency and Laplacian eigenvalues of the zero-divisor graph of a finite commutative von Neumann regular ring . We prove that is a generalized join of its induced subgraphs. Among the eigenvalues (respectively, Laplacian eigenvalues) of , exactly are the eigenvalues of a matrix obtained from the adjacency (respectively, Laplacian) matrix of -the zero-divisor graph of nontrivial idempotents in . We also determine the degree of each vertex in , hence the number of edges.
中文翻译:
冯诺依曼正则环的零除数图谱
零除数图交换环的是其顶点是非零零除数的图和两个顶点和相邻当且仅当. 我们研究了零除数图的邻接和拉普拉斯特征值有限交换冯诺依曼正则环的. 我们证明是其诱导子图的广义连接。之间的特征值(分别为拉普拉斯特征值), 确切地是从邻接(分别为拉普拉斯)矩阵获得的矩阵的特征值- 非平凡幂等的零除数图. 我们还确定了每个顶点的度数,因此是边数。