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Let $G$ be a graph on $[n]$ and $J_G$ be the binomial edge ideal of $G$ in the polynomial ring $S=\mathbb{K}[x_1,\ldots,x_n,y_1,\ldots,y_n]$. In this paper we investigate some topological properties of a poset associated to the minimal primary decomposition of $J_G$. We show that this poset admits some specific subposets which are contractible. This in turn, provides some interesting algebraic consequences. In particular, we characterize all graphs $G$ for which $\mathrm{depth}\hspace{1.2mm} S/J_G=4$., Comment: 13 pages, 3 figures, to appear in J. Algebra

In this paper, a question due to Heckenberger, Shareshian and Welker on racks in Heckenberger et al. (Trans Am Math Soc, 372:1407–1427, 2019) is positively answered. A rack is a set together with a self-distributive bijective binary operation. We show that the lattice of subracks of every finite rack is complemented. Moreover, we characterize finite modular lattices of subracks in terms of complements of subracks. Also, we introduce a certain class of racks including all finite groups with the conjugation operation, called G-racks, and we study some of their properties. In particular, we show that a finite G-rack has the homotopy type of a sphere. Further, we show that the lattice of subracks of an infinite rack is not necessarily complemented which gives an affirmative answer to the aforementioned question. Indeed, we show that the lattice of subracks of the set of rational numbers, as a dihedral rack, is not complemented. Finally, we show that being a Boolean algebra, pseudocomplemented and uniquely complemented as well as distributivity are equivalent for the lattice of subracks of a rack.

Let A be an abelian group. If for any x , y ∈ A we define x ⊳ y = 2 x − y , then ( A , ⊳ ) is a quandle called the Takasaki quandle and is denoted by T(A). In this article, we focus on the lattice ...

In this paper we prove the conjectured upper bound for Castelnuovo–Mumford regularity of binomial edge ideals posed in [23], in the case of chordal graphs. Indeed, we show that the regularity of any chordal graph G is bounded above by the number of maximal cliques of G, denoted by c(G). Moreover, we classify all chordal graphs G for which $$\mathcal {L}(G)=c(G)$$ , where $$\mathcal {L}(G)$$ is the sum of the lengths of longest induced paths of connected components of G. We call such graphs strong interval graphs. We show that the regularity of a strong interval graph G coincides with $$\mathcal {L}(G)$$ as well as c(G).

In this paper, we present some recent instances of applying algebraic tools from different categories, in the design of digital communications systems. More specifically, we present a structure based on the elements of a group ring for constructing and encoding QC-LDPC codes. As another instance, we present the construction of space-time block codes from the rings of twisted Laurent series which include some instances of crossed product and non-crossed product division algebras.

Let J G be the binomial edge ideal of a graph G. We characterize all graphs whose binomial edge ideals, as well as their initial ideals, have regularity 3. Consequently we characterize all graphs G such that J G is extremal Gorenstein. Indeed, these characterizations are consequences of an explicit formula we obtain for the regularity of the binomial edge ideal of the join product of two graphs. Finally, by using our regularity formula, we discuss some open problems in the literature. In particular we disprove a conjecture in [5] on the regularity of weakly closed graphs.

An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Let G be a bridgeless simple graph with minimum degree at least 3. Jackson and Yoshimoto (2007) showed that G has an even factor containing two arbitrary prescribed edges. They also proved that G has an even factor in which each component has order at least four. Moreover, Xiong, Lu and Han (2009) showed that for each pair of edges e1 and e2 of G, there is an even factor containing e1 and e2 in which each component containing neither e1 nor e2 has order at least four. In this paper we improve this result and prove that G has an even factor containing e1 and e2 such that each component has order at least four.

Low density parity check (LDPC) lattices are obtained from Construction D' and a family of nested binary LDPC codes. We consider an special case of these lattices with one binary LDPC code as underlying code. This special case of LDPC lattices can be obtained by lifting binary LDPC codes using Construction A lattices. The LDPC lattices were the first family of lattices which have efficient decoding in high dimensions. We employ the encoding and decoding of the LDPC lattices in a cooperative transmission framework. We establish two efficient shaping methods based on hypercube shaping and Voronoi shaping, to obtain LDPC lattice codes. Then, we propose the implementation of block Markov encoding for one-way and two-way relay networks using LDPC lattice codes. This entails owning an efficient method for decomposing full-rate codebook into lower rate codebooks. We apply different decomposition schemes for one-way and two-way relay channels which are the altered versions of the decomposition methods of low density lattice codes (LDLCs). Due to the lower complexity of the decoding for LDPC lattices comparing to LDLCs, the complexity of our schemes are significantly lower than the ones proposed for LDLCs. The efficiency of the proposed schemes are presented using simulation results., 30 pages, 2 figures

Ever since the introduction, skew inverse Laurent series rings have kept growing in importance, as researchers characterized their properties (such as Noetherianness, Armendarizness, McCoyness, etc.) in terms of intrinsic properties of the base ring and studied their relations with other fields of mathematics, as for example quantum mechanics theory. The goal of our paper is to study the primeness and semiprimeness of general skew inverse Laurent series rings R((x−1;σ,δ)), where R is an associative ring equipped with an automorphism σ and a σ-derivation δ.

Let R = k[x1,…, xn], where k is a field. The path ideal (of length t) of a directed graph G is the monomial ideal, denoted by It(G), whose generators correspond to the directed paths of length t in G. We determine all the graded Betti numbers of the path ideal of a directed rooted tree with respect to some graphical terms.

In this article, we consider a type of generalized group identity and extend some earlier results. For example, we show that, if D is a division ring with infinite center, then every subnormal subgroup of GLn(D) satisfying a generalized group identity over GLn(D) is central.

Given simple graphs G , H 1 , … , H n , where n = | V ( G ) | , the generalized corona, denoted G ∘ ˜ Λ i = 1 n H i , is the graph obtained by taking one copy of graphs G , H 1 , … , H n and joining the ith vertex of G to every vertex of H i . In this paper, we determine and study the characteristic, Laplacian and signless Laplacian polynomial of G ∘ ˜ Λ i = 1 n H i . This leads us to construct new pairs of cospectral, L-cospectral and Q-cospectral graphs. As an application, we give a simple proof for Csikvari's Lemma on eigenvalues of graphs.

In this paper, motivated by a question posed in \cite{AH}, we introduce strongly biconvex graphs as a subclass of weakly chordal and bipartite graphs. We give a linear time algorithm to find an induced matching for such graphs and we prove that this algorithm indeed gives a maximum induced matching. Applying this algorithm, we provide a strongly biconvex graph whose (monomial) edge ideal does not admit a unique extremal Betti number. Using this constructed graph, we provide an infinite family of the so-called closed graphs (also known as proper interval graphs) whose binomial edge ideals do not have a unique extremal Betti number. This, in particular, answers the aforementioned question in \cite{AH}., Comment: 19 pages, 2 figures

A rack is a set together with a self-distributive bijective binary operation. In this paper, we give a positive answer to a question due to Heckenberger, Shareshian and Welker. Indeed, we prove that the lattice of subracks of a rack is atomic. Further, by using the atoms, we associate certain quandles to racks. We also show that the lattice of subracks of a rack is isomorphic to the lattice of subracks of a quandle. Moreover, we show that the lattice of subracks of a rack is distributive if and only if its corresponding quandle is trivial. Finally, applying our corresponding quandles, we provide a coloring of certain knot diagrams., 15 pages, 10 figures

Hirano studied the quasi-Armendariz property of rings, and then this concept was generalized by some authors, defining quasi-Armendariz property for skew polynomial rings and monoid rings. In this article, we consider unified approach to the quasi-Armendariz property of skew power series rings and skew polynomial rings by considering the quasi-Armendariz condition in mixed extension ring [R; I][x; σ], introducing the concept of so-called (σ, I)-quasi Armendariz ring, where R is an associative ring equipped with an endomorphism σ and I is an σ-stable ideal of R. We study the ring-theoretical properties of (σ, I)-quasi Armendariz rings, and we obtain various necessary or sufficient conditions for a ring to be (σ, I)-quasi Armendariz. Constructing various examples, we classify how the (σ, I)-quasi Armendariz property behaves under various ring extensions. Furthermore, we show that a number of interesting properties of an (σ, I)-quasi Armendariz ring R such as reflexive and quasi-Baer property transfer to its...

We study unmixed and Cohen-Macaulay properties of the binomial edge ideal of some classes of graphs. We compute the depth of the binomial edge ideal of a generalized block graph. We also characterize all generalized block graphs whose binomial edge ideals are Cohen–Macaulay and unmixed. So that we generalize the results of Ene, Herzog, and Hibi on block graphs. Moreover, we study unmixedness and Cohen–Macaulayness of the binomial edge ideal of some graph products such as the join and corona of two graphs with respect to the original graphs.

Let G be a simple graph with the adjacency matrix A ( G ) . A well-known result of Cvetkovic and Gutman states that removing a pendant vertex and its neighbour, does not change the nullity of the graph. We generalize this theorem and some other theorems similar to it for an arbitrary eigenvalue of a graph. Also, for an arbitrary eigenvalue λ, we use a corresponding star set to delete some subgraphs and to determine the multiplicity of λ. We use star sets to find some Parter–Wiener vertices, in trees. We state our results for real symmetric matrices (and so for weighted graphs) and as a corollary we use them for simple graphs. Furthermore, we use these methods to obtain some results for λ = 0 , ± 1 .

We classify all complete uniform multipartite hypergraphs with respect to some algebraic properties, such as being (almost) complete intersection, Gorenstein, level, $l$-Cohen-Macaulay, $l$-Buchsbaum, unmixed, and satisfying Serre's condition $S_r$, via some combinatorial terms. Also, we prove that for a complete $s$-uniform $t$-partite hypergraph $H$, vertex decomposability, shellability, sequentially $S_r$ and sequentially Cohen-Macaulay properties coincide with the condition that $H$ has $t-1$ sides consisting of a single vertex. Moreover, we show that the latter condition occurs if and only if it is a chordal hypergraph., 11 pages, To appear in Communications in Algebra

We study a family of Cayley graphs on the group of n × n matrices M n ( F ) , where F is a finite field and n is a natural number, with the connection set of GL n ( F ) . We find that this graph is strongly regular only when n = 2 . We find diameter of this graph and we show that, every matrix in M n ( F ) is either invertible or sum of two invertible matrices. Moreover, we show that G M n ( F ) is class 1 if and only if char F = 2 . Finally, it is shown that for each graph G and each finite field F, G is an induced subgraph of Cay ( M n ( F ) , GL n ( F ) ) .

Let R = k[x1,…,xn], where k is a field. The path ideal (of length t ≥ 2) of a directed graph G is the monomial ideal, denoted by It(G), whose generators correspond to the directed paths of length t in G. Let Cn be an n-cycle. We show that R/It(Cn) is Sr if and only if it is Cohen-Macaulay or $\lceil \frac{n}{n-t-1}\rceil\geq r+3$. In addition, we prove that R/It(Cn) is Gorenstein if and only if n = t or 2t + 1. Also, we determine all ordinary and symbolic powers of It(Cn) which are Cohen-Macaulay. Finally, we prove that It(Cn) has a linear resolution if and only if t ≥ n − 2.

Let G be a graph and L ( G ) be the Laplacian matrix of G. In this paper, we explicitly determine the multiplicity of Laplacian eigenvalue 2 for any unicyclic graph containing a perfect matching.

In the present note, we continue to study zero-divisor properties of general skew inverse Laurent series rings R((x −1; σ, δ)), where R is an associative ring equipped with an automorphism σ and a σ-derivation δ. Extending the results of a large number of papers on the McCoy property for ordinary and skew polynomial rings, we study a version of the McCoy property for general skew inverse Laurent series extensions. We obtain some necessary or sufficient conditions for a ring to be (σ, δ)-SILS McCoy and prove that this property is preserved under a number of ring extensions. In particular, it passes to certain subrings of the (skew-)upper triangular matrix rings. In relation with this work and as an application of (σ, δ)-SILS McCoy rings, we investigate the interplay between the ring-theoretical properties of a general skew inverse Laurent series ring R((x −1; σ, δ)) and the graph-theoretical properties of its zero-divisor graph .

In this note, we continue to study zero-divisor properties of skew inverse Laurent series rings R((x −1; σ, δ)), where R is an associative ring equipped with an automorphism σ and a σ-derivation δ. We first introduce (σ, δ)-SILS Armendariz rings, a generalization of the standard Armendariz condition from ordinary polynomial ring R[x] to skew inverse Laurent series ring R((x −1; σ, δ)). We study the ring-theoretical properties of (σ, δ)-SILS Armendariz rings and using the properties of these rings, we characterize radicals of the skew inverse Laurent series ring R((x −1; σ, δ)), in terms of a (σ, δ)-SILS Armendariz ring R. We also prove that several properties transfer between R and R((x −1; σ, δ)), in case R is an σ-compatible (σ, δ)-SILS Armendariz ring.

Let D be a non-commutative division ring and M a maximal subgroup of GL n ( D ) ( n ⩾ 2 ). This paper continues the ongoing effort to show that the structure of maximal subgroups of GL n ( D ) is similar, in some sense, to the structure of GL n ( D ) . It is known that every locally soluble normal subgroup of GL n ( D ) is abelian. Here, among other results, we prove that if either (i) D is finite-dimensional over its center, or (ii) the center of D contains at least five elements and M is soluble-by-finite, or (iii) char D = 0 and M is (locally soluble)-by-finite, then every locally soluble normal subgroup of M is abelian.

Let Q G ( x ) = det ( xI - Q ( G ) ) = ∑ i = 0 n ( - 1 ) i ζ i x n - i be the characteristic polynomial of the signless Laplacian matrix of a graph G . Due to the nice properties of the signless Laplacian matrix, Q ( G ) , in comparison with the other matrices related to graphs, ζ -ordering, an ordering based on the coefficients of the signless Laplacian characteristic polynomial, is of interest. In this paper, using graph transformations, we establish some relations between the graph structure and its coefficients of the signless Laplacian characteristic polynomial. So, we express some results about ζ -ordering of graphs, focusing our attention to the unicyclic graphs. Finally, as an application of these results, we discuss the ordering of graphs based on their incidence energy.

This work is based on ideas of Ilic [A. Ilic, The energy of unitary Cayley graphs, Linear Algebra Appl. 431 (2009) 1881–1889] on the energy of unitary Cayley graph. For a finite commutative ring R with unity 1 ≠ 0 , the unitary Cayley graph of R is the Cayley graph whose vertex set is R and the edge set is { { a , b } : a , b ∈ R and a - b ∈ R × } , where R × is the group of units of R. We study the eigenvalues of the unitary Cayley graph of a finite commutative ring and some gcd-graphs and compute their energy. Moreover, we obtain the energy for the complement of unitary Cayley graphs.

Let D be a division ring and N be a subnormal subgroup of D*. In this paper we prove that if M is a nilpotent maximal subgroup of N, then M′ is abelian. If, furthermore every element of M is algebraic over Z(D) and M′ ⊈ F* or M/Z(M) or M′ is finitely generated, then M is abelian. The second main result of this paper concerns the subgroups of matrix groups; assume D is a noncommutative division ring, n is a natural number, N is a subnormal subgroup of GLn(D), and M is a maximal subgroup of N. We show that if M is locally finite over Z(D)*, then M is either absolutely irreducible or abelian.