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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.

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.

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

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 $$G$$ G be a graph of order $$n$$ n and size $$m$$ m . Suppose that $$f:V(G)\rightarrow {\mathbb {N}}$$ f : V ( G ) ? N is a function such that $$\sum _{v\in V(G)}f(v)=m+n$$ ? v ? V ( G ) f ( v ) = m + n . In this paper we provide a criterion for $$f$$ f -choosability of $$G$$ G . Using this criterion, it is shown that the choice number of the complete $$k$$ k -partite graph $$K_{2,2,\ldots ,2}$$ K 2 , 2 , ? , 2 is $$k$$ k , which is a well-known result due to Erdos, Rubin and Taylor. Among other results we study the $$f$$ f -choosability of the complete $$k$$ k -partite graphs with part sizes at most $$2$$ 2 , when $$f(v)\in \{k-1,k\}$$ f ( v ) ? { k - 1 , k } , for every vertex $$v\in V(G)$$ v ? V ( G ) .

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.

Let G be a graph and M be a matrix associated with G whose characteristic polynomial is M ( G , x ) = ∑ i = 0 n α i ( G ) x n − i . We say that the spectrum of G is determined by non-constant coefficients (simply M-SDNC), if for any graph H with a i ( H ) = a i ( G ) , 0 ⩽ i ⩽ n − 1 , then S p e c ( G ) = S p e c ( H ) (if M is the adjacency matrix or the Laplacian matrix of G , then G is called an A-SDNC graph or L-SDNC graph). In this paper, we study some properties of graphs which are A-SDNC or L-SDNC. Among other results, we prove that the path of order at least five is L-SDNC and moreover stars of order at least five are both A-SDNC and L-SDNC. Furthermore, we construct infinitely many trees which are not A-SDNC graphs. More precisely, we show that there are infinitely many pairs ( T , T ′ ) of trees such that A ( T , x ) − A ( T ′ , x ) = − 1 .

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 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.

An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Jackson and Yoshimoto showed that if G is a 3-edge-connected graph with |G| ≥ 5 and v is a vertex with degree 3, then G has an even factor F containing two given edges incident with v in which each component has order at least 5. We prove that this theorem is satisfied for each pair of adjacent edges. Also, we show that each 3-edge-connected graph has an even factor F containing two given edges e and f such that every component containing neither e nor f has order at least 5. But we construct infinitely many 3-edge-connected graphs that do not have an even factor F containing two arbitrary prescribed edges in which each component has order at least 5.

A matrix A over a field F is said to be an AJT matrix if there exists a vector x over F such that both x and Ax have no zero component. The Alon–Jaeger–Tarsi (AJT) conjecture states that if F is a finite field, with |F|≥4, and A is an element of GL n (F) , then A is an AJT matrix. In this paper we prove that every nonzero matrix over a field F, with |F|≥3 , is similar to an AJT matrix. Let AJTn (q) denote the set of n×n, invertible, AJT matrices over a field with q elements. It is shown that the following are equivalent for q≥3 : (i) AJTn (q)=GL n (q) ; (ii) every 2n×n matrix of the form (A∣B)t has a nowhere-zero vector in its image, where A,B are n×n, invertible, upper and lower triangular matrices, respectively; and (iii) AJTn (q) forms a semigroup.

The commuting graph of a ring R, denoted by Γ(R), is a graph of all whose vertices are noncentral elements of R, and 2 distinct vertices x and y are adjacent if and only if xy = yx. In this article we investigate some graph-theoretic properties of Γ(kG), where G is a finite group, k is a field, and 0 ≠ |G| ∈k. Among other results it is shown that if G is a finite nonabelian group and k is an algebraically closed field, then Γ(kG) is not connected if and only if |G| = 6 or 8. For an arbitrary field k, we prove that Γ(kG) is connected if G is a nonabelian finite simple group or G′ ≠ G″ and G″ ≠ 1.

Let D be a noncommutative finite dimensional F-central division algebra, and let N be a normal subgroup of GL n (D) with n ≥ 1. Given a maximal subgroup M of N, it is proved that either M contains a noncyclic free subgroup, or there exists an abelian subgroup A and a finite family of fields properly containing F with for all 1 ≤ i ≤ r such that M/A is finite if Char F = 0 and M/A is locally finite if Char F = p > 0, where .

Suppose that D is a division ring with center F and N is a non-central normal subgroup of GLn(D). In this paper we generalize some known results about maximal subgroups of GLn(D) to maximal subgroups of N. More precisely, we prove that if M is a maximal subgroup of N such that F[M] satisfies a polynomial identity and \(C_{M_{n}(D)}(M) \backslash F\) contains an algebraic element over F or \(C_{M_{n}(D)}(M)=F\) and either n ≥ 2 or n = 1 and M is not abelian, then [D : F] < ∞.

Let $R$ be a finite commutative ring. In this paper, we find the number of representations of a fixed member of $R$ to be the sum of $k$ units in $R$, and the sum of $k$ non-units, and as a sum of a unit and a non-unit. We prove that, if $\Z _2$ is not a quotient of $R$, then every $r\in R$ can be written as a sum of $k$ units, for each integer $k > 1$.

Let G be a mixed graph and L(G) be the Laplacian matrix of G. In this paper, the coefficients of the Laplacian characteristic polynomial ofG are studied. The first derivative of the characteristic polynomial of L(G) is explicitly expressed by means of Laplacian characteristic poly- nomials of its edge deleted subgraphs. As a consequence, it is shown that the Laplacian characteristic polynomial of a mixed graph is reconstructible from the collection of the Laplacian characteristic polynomials of its edge deleted subgraphs. Then, it is investigated how graph modifications affect the mixed Laplacian characteristic polynomial. Also, a connection between the Laplacian character- istic polynomial of a non-singular connected mixed graph and the signless Laplacian characteristic polynomial is provided, and it is used to establish a lower bound for the spectral radius of L(G). Finally, using Coates digraphs, the perturbation of the mixed Laplacian spectral radius under some graph transformations is discussed. V (G) and E(G), respectively. The incidence matrix of G is a {−1,0,1}-matrix, whose rows and columns are labelled by vertices and edges, respectively, and is denoted by M(G) = (mij)n×m. The entry mij = 1 if ej is an unoriented edge incident with vi, or if ej is an oriented edge with head vi, mij = −1 if ej is an oriented edge with tail vi, and mij = 0 otherwise. The Laplacian matrix of the mixed graph G is defined as L(G) = M(G)M(G) t , so that L(G) is positive semi-definite. Hence, its eigenvalues can be arranged as µ1(G) ≥ µ2(G) ≥ ··· ≥ µn(G) ≥ 0. One may see that, to obtain L(G), we only check that an edge is oriented or not, and do not care which vertex is the head and which one is the tail of an oriented edge. So, for each e ∈ E(G), we could consider a sign function which is denoted by sgn(e) and defined as sgn(e) = 1 if e is

Let D be a division ring with centre F. Assume that M is a maximal subgroup of GLn(D) (n≥1) such that Z(M) is algebraic over F. Group identities on M and polynomial identities on the F-linear hull F[M] are investigated. It is shown that if F[M] is a PI-algebra, then [D:F]n(D) and M is a maximal subgroup of N. If M satisfies a group identity, it is shown that M is abelian-by-finite.

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

We characterize all graphs whose binomial edge ideals have pure resolutions. Moreover, we introduce a new switching of graphs which does not change some algebraic invariants of graphs, and using this, we study the linear strand of the binomial edge ideals for some classes of graphs. Also, we pose a conjecture on the linear strand of such ideals for every graph., 10 pages, 8 figures, To appear in Collectanea Math

Let G be a graph and L(G) be the Laplacian matrix of G. In this article, we first point out that the sequence of the moduli of Laplacian coefficients ofG is log-concave and hence unimodal. Using this fact, we provide an upper bound for the partial sums of the Laplacian eigenvalues of G, based on coefficients of its Laplacian characteristic polynomial. We then obtain some lower bounds on the algebraic connectivity of G. Finally, we investigate the mode of such sequences. 1. Introduction. Throughout this paper, we consider simple undirected graphs having n vertices and m edges. For a given graph G, let V (G) and E(G) denote the vertex and the edge set of G, respectively. Let A(G) be the adjacency matrix of G. The Laplacian matrix and signless Laplacian matrix of G are defined as L(G) = D(G) − A(G) and Q(G) = D(G)+A(G), respectively, where D(G) is a diagonal matrix whose diagonal entries are vertex degrees of G. The eigenvalues of matrices L(G) and Q(G) are denoted by µ1(G) ≥ µ2(G) ≥ ··· ≥ µn(G) = 0 and �1(G) ≥ �2(G) ≥ ··· ≥ �n(G), respectively. As it is well-known, L(G) and Q(G) are positive semi-definite, and they have the same characteristic polynomial if and only if the graph G is bipartite. The second smallest eigenvalue of L(G), µn−1(G), is called the algebraic connectivity of G, and it is positive if and only if the graph G is connected. For bibliographies on the graph Laplacian, the reader is referred to (11).

We characterize all pairs of graphs $(G_1,G_2)$, for which the binomial edge ideal $J_{G_1,G_2}$ has linear relations. We show that $J_{G_1,G_2}$ has a linear resolution if and only if $G_1$ and $G_2$ are complete and one of them is just an edge. We also compute some of the graded Betti numbers of the binomial edge ideal of a pair of graphs with respect to some graphical terms. In particular, we show that for every pair of graphs $(G_1,G_2)$ with girth (i.e. the length of a shortest cycle in the graph) greater than 3, $\beta_{i,i+2}(J_{G_1,G_2})=0$, for all $i$. Moreover, we give a lower bound for the Castelnuovo-Mumford regularity of any binomial edge ideal $J_{G_1,G_2}$ and hence the ideal of adjacent $2$-minors of a generic matrix. We also obtain an upper bound for the regularity of $J_{G_1,G_2}$, if $G_1$ is complete and $G_2$ is a closed graph.

Let D be an infinite division ring, n a natural number and N a subnormal subgroup of GLn(D) such that n = 1 or the center of D contains at least five elements. This paper contains two main results. In the first one we prove that each nilpotent maximal subgroup of N is abelian; this generalizes the result in [R. Ebrahimian, J. Algebra 280 (2004) 244 - 248] (which asserts that each max- imal subgroup of GLn(D) is abelian) and a result in [M. Ramezan-Nassab, D. Kiani, J. Algebra 376 (2013) 1 - 9]. In the second one we show that a maximal subgroup of GLn(D) cannot be polycyclic-by-finite., Comment: 9 pages

We characterize all graphs whose binomial edge ideals have a linear resolution. Indeed, we show that complete graphs are the only graphs with this property. We also compute some graded components of the first Betti number of the binomial edge ideal of a graph with respect to the graphical terms. Finally, we give an upper bound for the Castelnuovo-Mumford regularity of the binomial edge ideal of a closed graph.

The proof of Theorem 3.3 in [Y. Zhang, X. Liu, B. Zhang, X. Yong, The lollipop graph is determined by its Q-spectrum, Discrete Math. 309 (2009) 3364–3369] is not correct. Actually, in the development of PA(G1)(λ), the authors missed several products into addition, which makes the rest of the proof invalid. Note that the statement of the theorem is true. Here, we give a correct proof.

We show that for the edge ideals of the graphs consisting of one cycle or two cycles of any length connected through a vertex, the arithmetical rank equals the projective dimension of the corresponding quotient ring.

In this paper a new class of vertex decomposable graphs are determined. Moreover, all shellable and sequentially Cohen-Macaulay cactus graphs (i.e., connected graphs in which each edge belongs to at most one cycle) are characterized.

Let D be an F-central division algebra of degree p r , p a prime. A set of criteria is given for D to be a crossed product in terms of irreducible soluble or abelian-by-finite subgroups of the multiplicative group D * of D. Using the Amitsur's classification of finite subgroups of D * and the Tits alternative, it is shown that D is a crossed product if and only if D * contains an irreducible soluble subgroup. Further criteria are also presented in terms of irreducible abelian-by-finite subgroups and irreducible subgroups satisfying a group identity. Using the above results, it is shown that if D * contains an irreducible finite subgroup, then D is a crossed product.

In this paper, we compute the projective dimension of the edge ideals of graphs consisting of some cycles and lines which are joint in a common vertex. Moreover, we show that for such graphs, the arithmetical rank equals the projective dimension. As an application, we can compute the arithmetical rank for some homogenous monomial ideals.