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Graphs associated to the set of irreducible character degrees of a finite group [Formula: see text] have been extensively studied as a way of understanding structure of the underlying group. … Graphs associated to the set of irreducible character degrees of a finite group [Formula: see text] have been extensively studied as a way of understanding structure of the underlying group. Another approach, proposed by Isaacs, is to study associated simplicial complexes, namely, the common divisor simplicial complex [Formula: see text] and the prime divisor simplicial complex [Formula: see text]. These complexes can be associated to any set of positive integers and this paper shows that they are homotopy equivalent. Further, considering these complexes associated to the set of irreducible character degrees, we give a bound on the rank of the fundamental group.
Coefficients of the cyclotomic polynomial can be interpreted topologically, as the torsion in the homology of a certain simplicial complex associated with the degree of the cyclotomic polynomial, which was … Coefficients of the cyclotomic polynomial can be interpreted topologically, as the torsion in the homology of a certain simplicial complex associated with the degree of the cyclotomic polynomial, which was studied by Musiker and Reiner. We answer a question posed by the two authors regarding homotopy type of certain subcomplexes of the associated simplicial complex when the degree of the cyclotomic polynomial is a product of three distinct primes.
Using discrete Morse theory for simplicial complexes we determine the homotopy type of ideal zerodivisor complex for finite rings and for rings with infinitely many maximal ideals. Using discrete Morse theory for simplicial complexes we determine the homotopy type of ideal zerodivisor complex for finite rings and for rings with infinitely many maximal ideals.
We study topology of the independence complexes of comaximal (hyper)graphs of commutative rings with identity. We show that the independence complex of comaximal hypergraph is contractible or homotopy equivalent to … We study topology of the independence complexes of comaximal (hyper)graphs of commutative rings with identity. We show that the independence complex of comaximal hypergraph is contractible or homotopy equivalent to a sphere, and that the independence complex of comaximal graph is almost always contractible.

Common Coauthors

Coauthor Papers Together
Zoran Z. Petrović 4
Aleksandra Kostić 2

Commonly Cited References

The general notion of a discriminant is as follows. Consider any function space $$\mathcal{F}$$ , finite dimensional or not, and some class of singularities S that the functions from $$\mathcal{F}$$ … The general notion of a discriminant is as follows. Consider any function space $$\mathcal{F}$$ , finite dimensional or not, and some class of singularities S that the functions from $$\mathcal{F}$$ can take at the points of the issue manifold. The corresponding discriminant variety ∑(S) ⊂ $$\mathcal{F}$$ is the space of all functions that have such singular points. For example, let $$\mathcal{F}$$ be the space of (real or complex) polynomials of the form 1 $${x^d} + {a_1}{x^{d - 1}} + \cdots + {a^{d,}}$$ and S = {a multiple root}.
The classical transportation problem is the study of the set of nonnegative matrices with prescribed nonnegative row and column sums. It is aesthetically satisfying and perhaps potentially useful to study … The classical transportation problem is the study of the set of nonnegative matrices with prescribed nonnegative row and column sums. It is aesthetically satisfying and perhaps potentially useful to study more general higher dimensional rectangular arrays whose sums on some subarrays are specified. We show how such problems can be rewritten as problems in homology theory. That translation explains the appearance of bipartite graphs in the study of the classical transportation problem. In our generalization, higher dimensional cell complexes occur. That is why the general problem requires a substantial independent investigation of simplicial geometry, the name given to the class of theorems on the geometry of a cell complex which depend on a particular cellular decomposition. The topological invariants of the complex are means, not ends. Thus simplicial geometry attempts to do for complexes what graph theory does for graphs. The dual title of this paper indicates that we shall spend as much time studying simplicial geometry for its own sake as applying the results to transportation problems. Our results include formulas for inverting the boundary operator of an acyclic cell complex, and some information on the number of such subcomplexes of a given complex.
Let R be a commutative ring with unity.We define a semi-simplicial abelian group based on the structure of the semigroup of ideals of R and investigate various properties of the … Let R be a commutative ring with unity.We define a semi-simplicial abelian group based on the structure of the semigroup of ideals of R and investigate various properties of the homology groups of the associated chain complex.
Prerequisites.- I * Knots and Knot Types.- 1. Definition of a knot.- 2. Tame versus wild knots.- 3. Knot projections.- 4. Isotopy type, amphicheiral and invertible knots.- II * The … Prerequisites.- I * Knots and Knot Types.- 1. Definition of a knot.- 2. Tame versus wild knots.- 3. Knot projections.- 4. Isotopy type, amphicheiral and invertible knots.- II * The Fundamental Group.- 1. Paths and loops.- 2. Classes of paths and loops.- 3. Change of basepoint.- 4. Induced homomorphisms of fundamental groups.- 5. Fundamental group of the circle.- III * The Free Groups.- 1. The free group F[A].- 2. Reduced words.- 3. Free groups.- IV * Presentation of Groups.- 1. Development of the presentation concept.- 2. Presentations and presentation types.- 3. The Tietze theorem.- 4. Word subgroups and the associated homomorphisms.- 5. Free abelian groups.- V * Calculation of Fundamental Groups.- 1. Retractions and deformations.- 2. Homotopy type.- 3. The van Kampen theorem.- VI * Presentation of a Knot Group.- 1. The over and under presentations.- 2. The over and under presentations, continued.- 3. The Wirtinger presentation.- 4. Examples of presentations.- 5. Existence of nontrivial knot types.- VII * The Free Calculus and the Elementary Ideals.- 1. The group ring.- 2. The free calculus.- 3. The Alexander matrix.- 4. The elementary ideals.- VIII * The Knot Polynomials.- 1. The abelianized knot group.- 2. The group ring of an infinite cyclic group.- 3. The knot polynomials.- 4. Knot types and knot polynomials.- IX * Characteristic Properties of the Knot Polynomials.- 1. Operation of the trivialize.- 2. Conjugation.- 3. Dual presentations.- Appendix I. Differentiable Knots are Tame.- Appendix II. Categories and groupoids.- Appendix III. Proof of the van Kampen theorem.- Guide to the Literature.
Elements of Algebraic Topology provides the most concrete approach to the subject. With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to … Elements of Algebraic Topology provides the most concrete approach to the subject. With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to classical theorems of point-set topology, this book is perfect for comunicating complex topics and the fun nature of algebraic topology for beginners.
In this paper, a new kind of graph on a commutative ring R with identity, namely the co-maximal ideal graph is defined and studied. We use [Formula: see text] to … In this paper, a new kind of graph on a commutative ring R with identity, namely the co-maximal ideal graph is defined and studied. We use [Formula: see text] to denote this graph, with its vertices the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I 1 and I 2 are adjacent if and only if I 1 + I 2 = R. We show some properties of this graph. For example, this graph is a simple, connected graph with diameter less than or equal to three, and both the clique number and the chromatic number of the graph are equal to the number of maximal ideals of the ring R.
Abstract In this paper we investigate the center, radius and girth of comaximal graphs of commutative rings. We also provide some counterexamples to the results concerning the relation between isomorphisms … Abstract In this paper we investigate the center, radius and girth of comaximal graphs of commutative rings. We also provide some counterexamples to the results concerning the relation between isomorphisms of comaximal graphs and the rings in question. In addition, we investigate the relation between the comaximal graph of a ring and its subrings of a certain type.
Let R be a ring with unity and I(R)* be the set of all nontrivial left ideals of R. The intersection graph of ideals of R, denoted by G(R), is … Let R be a ring with unity and I(R)* be the set of all nontrivial left ideals of R. The intersection graph of ideals of R, denoted by G(R), is a graph with the vertex set I(R)* and two distinct vertices I and J are adjacent if and only if I ∩ J ≠ 0. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose intersection graphs of ideals are not connected. Also we determine all rings whose clique number of the intersection graphs of ideals is finite. Among other results, it is shown that for a ring R, if the clique number of G(R) is finite, then the chromatic number is finite and if R is a reduced ring, then both are equal.
In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of … In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid structure on the set of faces. This theory was later extended by Brown to a larger class of monoids called left regular bands. In both cases, the representation theory of these monoids played a prominent role. In particular, it was used to compute the spectrum of the transition operators of the Markov chains and to prove diagonalizability of the transition operators. In this paper, we establish a close connection between algebraic and combinatorial invariants of a left regular band: we show that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. For instance, we show that the global dimension of these algebras is bounded above by the Leray number of the associated order complex. Conversely, we associate to every flag complex a left regular band whose algebra has global dimension precisely the Leray number of the flag complex.
Abstract. We interpret the coefficients of the cyclotomic polynomial in terms of simplicial homology. Abstract. We interpret the coefficients of the cyclotomic polynomial in terms of simplicial homology.
These lecture notes for the IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004) provide an overview of poset topology. These notes include introductory material, as well as recent … These lecture notes for the IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004) provide an overview of poset topology. These notes include introductory material, as well as recent developments and open problems. Some of the topics covered are: subspace arrangements, graph complexes, group actions on poset homology, shellability, recursive techniques, and fiber theorems.
Gelfand, Retakh, Serconek and Wilson, in \cite{GRSW}, defined a graded algebra $A_\Gamma$ attached to any finite ranked poset $\Gamma$ - a generalization of the universal algebra of pseudo-roots of noncommutative … Gelfand, Retakh, Serconek and Wilson, in \cite{GRSW}, defined a graded algebra $A_\Gamma$ attached to any finite ranked poset $\Gamma$ - a generalization of the universal algebra of pseudo-roots of noncommutative polynomials. This algebra has since come to be known as the splitting algebra of $\Gamma$. The splitting algebra has a secondary filtration related to the rank function on the poset and the associated graded algebra is denoted here by $A'_\Gamma$. We calculate the cohomology algebra (and coalgebra) of $A'_\Gamma$ explicitly. As a corollary to this calculation we have a proof that $A'_\Gamma$ is Koszul (respectively quadratic) if and only if $\Gamma$ is Cohen-Macaulay (respectively uniform). We show by example that the cohomology algebra (resp. coalgebra) of $A_\Gamma$ may be strictly smaller that the cohomology algebra (resp. coalgebra) of $A'_\Gamma$.
Let Alt _k be the alternating group of degree k . In this paper we prove that the order complex of the coset poset of Alt _k is non-contractible for … Let Alt _k be the alternating group of degree k . In this paper we prove that the order complex of the coset poset of Alt _k is non-contractible for a big family of k\in {\mathbb N} , including the numbers of the form k=p+m where m\in \{3,\ldots,35\} and p> k/2 . In order to prove this result, we show that P_G(-1) does not vanish, where P_G(s) is the Dirichlet polynomial associated to the group G . Moreover, we extend the result to some monolithic primitive groups whose socle is a direct product of alternating groups.