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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).

A graph $$G=(V(G), E(G))$$ is supereulerian if it has a spanning Eulerian subgraph. Let $$\ell (G)$$ be the maximum number of edges of spanning Eulerian subgraphs of a graph G. Motivated by a conjecture due to Catlin on supereulerian graphs, it was shown that if G is an r-regular supereulerian graph, then $$\ell (G)\ge \frac{2}{3}|E(G)|$$ when $$r\ne 5$$ , and $$\ell (G)> \frac{3}{5}|E(G)|$$ when $$r=5$$ . In this paper we improve the coefficient and prove that if G is a 5-regular supereulerian graph, then $$\ell (G)\ge \frac{19}{30}|E(G)|+\frac{4}{3}$$ . For this, we first show that each graph G with maximum degree at most 3 has a matching with at least $$\frac{2}{7}|E(G)|$$ edges and this bound is sharp. Moreover, we show that Catlin’s conjecture holds for claw-free graphs having no vertex of degree 4. Indeed, Catlin’s conjecture does not hold for claw-free graphs in general.

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.

In this paper we introduce the concept of clique disjoint edge sets in graphs. Then, for a graph G, we define the invariant η ( G ) as the maximum size of a clique disjoint edge set in G. We show that the regularity of the binomial edge ideal of G is bounded above by η ( G ) . This, in particular, settles a conjecture on the regularity of binomial edge ideals in full generality.

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.

We prove a conjectured upper bound for the Castelnuovo-Mumford regularity of binomial edge ideals of graphs, due to Matsuda and Murai. Indeed, we prove that reg ( J G ) ? n - 1 for any graph G with n vertices, which is not a path.

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

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...

Let $M=[m_{ij}]$ be an $n\times m$ real matrix, $\rho$ be a nonzero real number, and $A$ be a symmetric real matrix. We denote by $D(M)$ the $n\times n$ diagonal matrix $diag(\sum_{j=1}^{m}m_{1j},\ldots,\sum_{j=1}^{m}m_{nj})$ and denote by $L_{A}^{\rho}$ the generalized Laplacian matrix $D(A)-\rho A$. A well-known result of Grone et al. states that by connecting one of the end-vertices of $P_{3}$ to an arbitrary vertex of a graph, does not change the multiplicity of Laplacian eigenvalue $1$. We extend this theorem and some other results for a given generalized Laplacian eigenvalue $\mu$. Furthermore, we give two proofs for a conjecture by Saito and Woei on the relation between the multiplicity of some Laplacian eigenvalues and pendant paths., Comment: 13 pages, 9 figures, two corrected typos in Lemma 20, to appear in Linear Algebra and its Applications

Let $R=k[x_{1},\ldots,x_{n}]$, where $k$ is a field. The path ideal (of length $t\geq 2$) of a directed graph $G$ is the monomial ideal, denoted by $I_{t}(G)$, whose generators correspond to the directed paths of length $t$ in $G$. Let $\Gamma$ be a directed rooted tree. We characterize all such trees whose path ideals are unmixed and Cohen-Macaulay. Moreover, we show that $R/I_{t}(\Gamma)$ is Gorenstein if and only if the Stanley-Reisner simplicial complex of $I_{t}(\Gamma)$ is a matroid., Comment: 10 pages, 4 figures, to appear in Algebra Colloquium. arXiv admin note: text overlap with arXiv:0708.3111 by other authors