The Experts below are selected from a list of 15 Experts worldwide ranked by ideXlab platform

Jinming Zhou - One of the best experts on this subject based on the ideXlab platform.

  • automorphisms of the zero divisor graph of the full matrix ring
    Linear & Multilinear Algebra, 2017
    Co-Authors: Jinming Zhou, Dein Wong
    Abstract:

    The zero-divisor graph of a non-commutative ring R, written as , is a directed graph with Vertex set of all non-zero zero-divisors of R, and there is a directed edge from a Vertex x to a Distinct Vertex y if and only if . Let M(n, q) (resp., T(n, q)) be the ring of all matrices (resp., upper triangular matrices) over a finite field . Recently, Wang (Linear Algebra Appl. 2015;465:214–220) determined the automorphisms of the zero-divisor graph of T(n, q). In this paper, we determine the automorphisms of , extending the result due to L. Wang from T(n, q) to M(n, q). Since the case is trivial, and the case has been examined in Ma et al. (J. Korean Math. Soc. 2016;53:519–532), we just determine the automorphisms of in the case. We show that a bijective map on is an automorphism of if and only if there exist invertible matrices and a such that for any , where , and depend on A, and .

  • automorphisms of the zero divisor graph over 2 2 matrices
    Journal of The Korean Mathematical Society, 2016
    Co-Authors: Dengyin Wang, Jinming Zhou
    Abstract:

    The zero-divisor graph of a noncommutative ring R, denoted by ( R), is a graph whose vertices are nonzero zero-divisors of R, and there is a directed edge from a Vertex x to a Distinct Vertex y if and only if xy = 0. Let R = M2(Fq) be the 2×2 matrix ring over a finite field Fq. In this article, we investigate the automorphism group of ( R).

Dein Wong - One of the best experts on this subject based on the ideXlab platform.

  • automorphisms of the zero divisor graph of the full matrix ring
    Linear & Multilinear Algebra, 2017
    Co-Authors: Jinming Zhou, Dein Wong
    Abstract:

    The zero-divisor graph of a non-commutative ring R, written as , is a directed graph with Vertex set of all non-zero zero-divisors of R, and there is a directed edge from a Vertex x to a Distinct Vertex y if and only if . Let M(n, q) (resp., T(n, q)) be the ring of all matrices (resp., upper triangular matrices) over a finite field . Recently, Wang (Linear Algebra Appl. 2015;465:214–220) determined the automorphisms of the zero-divisor graph of T(n, q). In this paper, we determine the automorphisms of , extending the result due to L. Wang from T(n, q) to M(n, q). Since the case is trivial, and the case has been examined in Ma et al. (J. Korean Math. Soc. 2016;53:519–532), we just determine the automorphisms of in the case. We show that a bijective map on is an automorphism of if and only if there exist invertible matrices and a such that for any , where , and depend on A, and .

  • automorphism group of an ideal relation graph over a matrix ring
    Linear & Multilinear Algebra, 2016
    Co-Authors: Dein Wong
    Abstract:

    The ideal-relation graph over a ring , written as , is a directed graph which has as Vertex set and there is a directed edge from a Vertex to a Distinct Vertex if and only if the (two-sided) ideal of generated by is properly contained in the ideal generated by . In this paper, the automorphisms of are characterized when is the ring of all upper triangular matrices over a finite field.

Dengyin Wang - One of the best experts on this subject based on the ideXlab platform.

  • automorphisms of the zero divisor graph over 2 2 matrices
    Journal of The Korean Mathematical Society, 2016
    Co-Authors: Dengyin Wang, Jinming Zhou
    Abstract:

    The zero-divisor graph of a noncommutative ring R, denoted by ( R), is a graph whose vertices are nonzero zero-divisors of R, and there is a directed edge from a Vertex x to a Distinct Vertex y if and only if xy = 0. Let R = M2(Fq) be the 2×2 matrix ring over a finite field Fq. In this article, we investigate the automorphism group of ( R).

Sisson Benjamin - One of the best experts on this subject based on the ideXlab platform.

  • Efficient Reassembling of Three-Regular Planar Graphs
    2018
    Co-Authors: Kfoury Assaf, Sisson Benjamin
    Abstract:

    A reassembling of a simple graph G = (V,E) is an abstraction of a problem arising in earlier studies of network analysis. There are several equivalent definitions of graph reassembling; in this report we use a definition which makes it closest to the notion of graph carving. A reassembling is a rooted binary tree whose nodes are subsets of V and whose leaf nodes are singleton sets, with each of the latter containing a Distinct Vertex of G. The parent of two nodes in the reassembling is the union of the two children's Vertex sets. The root node of the reassembling is the full set V. The edge-boundary degree of a node in the reassembling is the number of edges in G that connect vertices in the node's set to vertices not in the node's set. A reassembling's alpha-measure is the largest edge-boundary degree of any node in the reassembling. A reassembling of G is alpha-optimal if its alpha-measure is the minimum among all alpha-measures of G's reassemblings. The problem of finding an alpha-optimal reassembling of a simple graph in general was already shown to be NP-hard. In this report we present an algorithm which, given a 3-regular plane graph G = (V,E) as input, returns a reassembling of G with an alpha-measure independent of n (number of vertices in G) and upper-bounded by 2k, where k is the edge-outerplanarity of G. (Edge-outerplanarity is Distinct but closely related to the usual notion of outerplanarity; as with outerplanarity, for a fixed edge-outerplanarity k, the number n of vertices can be arbitrarily large.) Our algorithm runs in time linear in n. Moreover, we construct a class of $3$-regular plane graphs for which this alpha-measure is optimal, by proving that 2k is the lower bound on the alpha-measure of any reassembling of a graph in that class.Comment: 49 pages, 25 figures, 15 reference

Kfoury Assaf - One of the best experts on this subject based on the ideXlab platform.

  • Efficient Reassembling of Three-Regular Planar Graphs
    2018
    Co-Authors: Kfoury Assaf, Sisson Benjamin
    Abstract:

    A reassembling of a simple graph G = (V,E) is an abstraction of a problem arising in earlier studies of network analysis. There are several equivalent definitions of graph reassembling; in this report we use a definition which makes it closest to the notion of graph carving. A reassembling is a rooted binary tree whose nodes are subsets of V and whose leaf nodes are singleton sets, with each of the latter containing a Distinct Vertex of G. The parent of two nodes in the reassembling is the union of the two children's Vertex sets. The root node of the reassembling is the full set V. The edge-boundary degree of a node in the reassembling is the number of edges in G that connect vertices in the node's set to vertices not in the node's set. A reassembling's alpha-measure is the largest edge-boundary degree of any node in the reassembling. A reassembling of G is alpha-optimal if its alpha-measure is the minimum among all alpha-measures of G's reassemblings. The problem of finding an alpha-optimal reassembling of a simple graph in general was already shown to be NP-hard. In this report we present an algorithm which, given a 3-regular plane graph G = (V,E) as input, returns a reassembling of G with an alpha-measure independent of n (number of vertices in G) and upper-bounded by 2k, where k is the edge-outerplanarity of G. (Edge-outerplanarity is Distinct but closely related to the usual notion of outerplanarity; as with outerplanarity, for a fixed edge-outerplanarity k, the number n of vertices can be arbitrarily large.) Our algorithm runs in time linear in n. Moreover, we construct a class of $3$-regular plane graphs for which this alpha-measure is optimal, by proving that 2k is the lower bound on the alpha-measure of any reassembling of a graph in that class.Comment: 49 pages, 25 figures, 15 reference