Some Properties of a Cayley Graph of a Commutative Ring

Let R be a commutative ring with unity and R ⁺, U(R), and Z*(R) be the additive group, the set of unit elements, and the set of all nonzero zero-divisors of R, respectively. We denote by ?𝔸𝕐(R) and G _R , the Cayley graph Cay(R ⁺, Z*(R)) and the unitary Cayley graph Cay(R ⁺, U(R)), respectively. For an Artinian ring R, Akhtar et al. (2009) studied G _R . In this article, we study ?𝔸𝕐(R) and determine the clique number, chromatic number, edge chromatic number, domination number, and the girth of ?𝔸𝕐(R). We also characterize all rings R whose ?𝔸𝕐(R) is planar. Moreover, we determine all finite rings R whose ?𝔸𝕐(R) is strongly regular. We prove that ?𝔸𝕐(R) is strongly regular if and only if it is edge transitive. As a consequence, we characterize all finite rings R for which G _R is a strongly regular graph.     

Nonzero Component Graph of a Finite Dimensional Vector Space

In this article, we introduce a graph structure, called a nonzero component graph on finite dimensional vector spaces. We show that the graph is connected and find its domination number and independence number. We also study the inter-relationship between vector space isomorphisms and graph isomorphisms, and it is shown that two graphs are isomorphic if and only if the corresponding vector spaces are so. Finally, we determine the degree of each vertex in case the base field is finite.     

Orthogonal Groups 𝒪(n) over GF(2) as Automorphisms

Let (V, Q) be a quadratic vector space over a fixed field. Orthogonal group 𝒪(V, Q) is defined as automorphisms on (V, Q). If Q = I, it is 𝒪(V, I) = 𝒪(n). There is a nice result that 𝒪(n) ? Aut(𝔬(n)) over ? or ?, where 𝔬(n) is the Lie algebra of n × n alternating matrices over the field. How about another field The answer is “Yes” if it is GF(2). We show it explicitly with the combinatorial basis ?. This is a verification of Steinberg's main result in 1961, that is, Aut(𝔬(n)) is simple over the square field, with a nonsimple exception Aut(𝔬(5)) ? 𝒪(5) ? 𝔖₆.     

Non-zero component union graph of a finite-dimensional vector space

In this paper, we introduce a graph structure, called non-zero component union graph on finite-dimensional vector spaces. We show that the graph is connected and find its domination number, clique number and chromatic number. It is shown that two non-zero component union graphs are isomorphic if and only if the base vector spaces are isomorphic. In case of finite fields, we study the edge-connectivity and condition under which the graph is Eulerian. Moreover, we provide a lower bound for the independence number of the graph. Finally, we come up with a structural characterization of non-zero component union graph.     

THE ADJOINT SEMIGROUP OF A RING

Abstract

We classify cross-sections of the ? and ? Green's relations on the finite symmetric inverse semigroup ?𝒮 _n , determine which of them are isomorphic, and study their disposition with respect to the ?-cross-sections of ?𝒮 _n .     

EXAMPLES OF FCR-ALGEBRAS

Abstract

Let R be a prime ring and 𝒰(R) its group of units. We prove that if 𝒰(R) satisfies a group identity and 𝒰(R) generates R,then either R is a domain or R is isomorphic to the algebra of n × n matrices over a finite field of order d. Moreover the integers n and d depend only on the group identity satisfed by 𝒰(R). This result has been recently proved by C. H. Liu and T. K. Lee (Liu,C. H.; Lee,T. K. Group identities and prime rings generated by units. Comm. Algebra (to appear)) and here we present a new different proof.     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

On resolvable primal spaces

A topological space is called resolvable if it is a union of two disjoint dense subsets, and is n-resolvable if it is a union of n mutually disjoint dense subsets. Clearly a resolvable space has no isolated points. If f is a selfmap on X, the sets A?X with f (A)?A are the closed sets of an Alexandroff topology called the primal topology 𝒫(f ) associated with f. We investigate resolvability for primal spaces (X, 𝒫(f)). Our main result is that an Alexandroff space is resolvable if and only if it has no isolated points. Moreover, n-resolvability and other related concepts are investigated for primal spaces.     

Vertex Poisson Algebras Associated with Courant Algebroids

We study relationships between vertex Poisson algebras and Courant algebroids. For any ?-graded vertex Poisson algebra A = ? _n∈? A _(n), we show that A ₍₁₎ is a Courant A ₍₀₎-algebroid. On the other hand, for any Courant 𝒜-algebroid ?, we construct an ?-graded vertex Poisson algebra A = ? _n∈? A _(n) such that A ₍₀₎ is 𝒜 and the Courant 𝒜-algebroid A ₍₁₎ is isomorphic to ? as a Courant 𝒜-algebroid.     

Matroidal families of finite connected nonhomeomorphic graphs exist

A matroidal family is a nonempty set ? of connected finite graphs such that for every arbitrary finite graph G the edge sets of the subgraphs of G which are isomorphic to an element of ? form a matroid on the edge set of G. In the present paper the question whether there are any matroidal families besides the four previously described by Simões-Pereira is answered affirmatively. It is shown that for every natural number n ? 2 there is a matroidal family that contains the complete graph with n vertices. For n = 4 this settles Simões-Pereira's conjecture that there is a matroidal family containing all wheels.     

Orthogonal latin square graphs

An orthogonal latin square graph (OLSG) is one in which the vertices are latin squares of the same order and on the same symbols, and two vertices are adjacent if and only if the latin squares are orthogonal. If G is an arbitrary finite graph, we say that G is realizable as an OLSG if there is an OLSG isomorphic to G. The spectrum of G [Spec(G)] is defined as the set of all integers n that there is a realization of G by latin squares of order n. The two basic theorems proved here are (1) every graph is realizable and (2) for any graph G, Spec G contains all but a finite set of integers. A number of examples are given that point to a number of wide open questions. An example of such a question is how to classify the graphs for which a given n lies in the spectrum.     

Random Graph Coverings I: General Theory and Graph Connectivity

In this paper we describe a simple model for random graphs that have an n-fold covering map onto a fixed finite base graph. Roughly, given a base graph G and an integer n, we form a random graph by replacing each vertex of G by a set of n vertices, and joining these sets by random matchings whenever the corresponding vertices are adjacent in G. The resulting graph covers the original graph in the sense that the two are locally isomorphic. We suggest possible applications of the model, such as constructing graphs with extremal properties in a more controlled fashion than offered by the standard random models, and also "randomizing" given graphs. The main specific result that we prove here (Theorem 1) is that if is the smallest vertex degree in G, then almost all n-covers of G are -connected. In subsequent papers we will address other graph properties, such as girth, expansion and chromatic number. Received June 21, 1999/Revised November 16, 2000 RID="*" ID="*" Work supported in part by grants from the Israel Academy of Aciences and the Binational Israel-US Science Foundation.     

Finite subgraphs of uncountably chromatic graphs

It is consistent that for every function f:ω → ω there is a graph with size and chromatic number ?₁ in which every n‐chromatic subgraph contains at least f(n) vertices (n ≥ 3). This solves a $ 250 problem of Erd?s. It is consistent that there is a graph X with Chr(X)=|X|=?₁ such that if Y is a graph all whose finite subgraphs occur in X then Chr(Y)≤?₂ (so the Taylor conjecture may fail). It is also consistent that if X is a graph with chromatic number at least ?₂ then for every cardinal λ there exists a graph Y with Chr(Y)≥λ all whose finite subgraphs are induced subgraphs of X. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 28–38, 2005     

On symplectic polar spaces over non-perfect fields of characteristic 2

Given a field 𝕂 of characteristic 2 and an integer n ≥ 2, let W(2n ? 1, 𝕂) be the symplectic polar space defined in PG(2n ? 1, 𝕂) by a non-degenerate alternating form of V(2n, 𝕂) and let Q(2n, 𝕂) be the quadric of PG(2n, 𝕂) associated to a non-singular quadratic form of Witt index n. In the literature it is often claimed that W(2n ? 1, 𝕂) ? Q(2n, 𝕂). This is true when 𝕂 is perfect, but false otherwise. In this article, we modify the previous claim in order to obtain a statement that is correct for any field of characteristic 2. Explicitly, we prove that W(2n ? 1, 𝕂) is indeed isomorphic to a non-singular quadric Q, but when 𝕂 is non-perfect the nucleus of Q has vector dimension greater than 1. So, in this case, Q(2n, 𝕂) is a proper subgeometry of W(2n ? 1, 𝕂). We show that, in spite of this fact, W(2n ? 1, 𝕂) can be embedded in Q(2n, 𝕂) as a subgeometry and that this embedding induces a full embedding of the dual DW(2n ? 1, 𝕂) of W(2n ? 1, 𝕂) into the dual DQ(2n, 𝕂) of Q(2n, 𝕂).     

Graphs with given group and given constant link

A graph L is called a link graph if there is a graph G such that for each vertex of G its neighbors induce a subgraph isomorphic to L. Such a G is said to have constant link .L Sabidussi proved that for any finite group F and any n ? 3 there are infinitely many n-regular connected graphs G with AutG ? Γ. We will prove a stronger result: For any finite group Γ and any link graph L with at least one isolated vertex and at least three vertices there are infinitely many connected graphs G with constant link L and AutG ? Γ.     

Exceptional Pairs in Hereditary Categories

We study the category 𝒞(X, Y) generated by an exceptional pair (X, Y) in a hereditary category ?. If r = dim _k Hom(X, Y) ≥ 1 we show that there are exactly 3 possible types for 𝒞(X, Y), all derived equivalent to the category of finite dimensional modules mod(H _r ) over the r-Kronecker algebra H _r . In general 𝒞(X, Y) will not be equivalent to a module category. More specifically, if ? is the category of coherent sheaves over a weighted projective line 𝕏, then 𝒞(X, Y) is equivalent to the category of coherent sheaves on the projective line ?¹ or to mod(H _r ) and, if 𝕏 is wild, then every r ≥ 1 can occur in this way.     

Hilbert Scheme of a Pair of Codimension Two Linear Subspaces

We study the component H _n of the Hilbert scheme whose general point parameterizes a pair of codimension two linear subspaces in ? ⁿ for n ≥ 3. We show that H _n is smooth and isomorphic to the blow-up of the symmetric square of 𝔾(n ? 2, n) along the diagonal. Further H _n intersects only one other component in the full Hilbert scheme, transversely. We determine the stable base locus decomposition of its effective cone and give modular interpretations of the corresponding models, hence conclude that H _n is a Mori dream space.     

On the average genus of the random graph

The generalized Petersen graph GP (n, k), n ≤ 3, 1 ≥ k < n/2 is a cubic graph with vertex-set {u_j; i ? Z_n} ∪ {v_j; i ? Z_n}, and edge-set {u_iu_i, u_iv_i, v_iv_{i+k, i?}Z_n}. In the paper we prove that (i) GP(n, k) is a Cayley graph if and only if k² ? 1 (mod n); and (ii) GP(n, k) is a vertex-transitive graph that is not a Cayley graph if and only if k² ? -1 (mod n) or (n, k) = (10, 2), the exceptional graph being isomorphic to the 1-skeleton of the dodecahedon. The proof of (i) is based on the classification of orientable regular embeddings of the n-dipole, the graph consisting of two vertices and n parallel edges, while (ii) follows immediately from (i) and a result of R. Frucht, J.E. Graver, and M.E. Watkins [“The Groups of the Generalized Petersen Graphs,” Proceedings of the Cambridge Philosophical Society, Vol. 70 (1971), pp. 211-218]. © 1995 John Wiley & Sons, Inc.     

Orthogonal one-factorization graphs

An orthogonal one-factorization graph (OOFG) is a graph in which the vertices are one-factorizations of some underlying graph H, and two vertices are adjacent if and only if the one-factorizations are orthogonal. An arbitrary finite graph, G, is realizable if there is an OOFG isomorphic to G. We show that every finite graph is realizable as an OOFG with underlying graph K_n for some n. We also discuss some special cases.     

Automorphism Group of the Rank-decreasing Graph Over the Semigroup of Upper Triangular Matrices

Let F_q be a finite field with q elements, n ≥ 2 a positive integer, and T(n, q) the semigroup of all n × n upper triangular matrices over F_q. The rank-decreasing graph 𝕋 of T(n, q) is a directed graph which has T(n, q) as vertex set, and there is a directed edge from A ∈ T(n, q) to B ∈ T(n, q) if and only if r(AB) < r(B). The zero-divisor graph 𝒯 of T(n, q), with vertex set of all nonzero zero-divisors of T(n, q) and there is a directed edge from a vertex A to a vertex B if and only if AB = 0, can be viewed as a subgraph of 𝕋. In [16 Wang, L. (2015). A note on automorphisms of the zero-divisor graph of upper triangular matrices. Lin. Alg. Appl. 465:214–220.[Crossref], [Web of Science ®] , [Google Scholar]], L. Wang has determined the automorphisms of the zero-divisor graph 𝒯 of T(n, q). In this article, by applying the main result of [17 Wong, D., Ma, X., Zhou, J. (2014). The group of automorphisms of a zero-divisor graph based on rank one upper triangular matrices. Lin. Alg. Appl. 460:242–258.[Crossref], [Web of Science ®] , [Google Scholar]] we determine the automorphisms of the rank-decreasing graph 𝕋 of T(n, q).