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We study commutative Schur rings over the symplectic groups Sp (n , 2) that contain the class C of symplectic transvections. We find the possible partitions of C determined by the Schur ring. We show how this restricts the possibilities for multiplicity one subgroups of Sp (n , 2). [ABSTRACT FROM AUTHOR]

The minimum number of idempotent generators is calculated for an incidence algebra of a finite poset over a commutative ring. This quantity equals either $\lceil \log _2 n\rceil $ or $\lceil \log _2 n\rceil +1$ , where n is the cardinality of the poset. The two cases are separated in terms of the embedding of the Hasse diagram of the poset into the complement of the hypercube graph. [ABSTRACT FROM AUTHOR]

This paper centers around one of several generalizations of primary ideals. Which is an intermediate class between S-primary ideals and quasi S-primary ideals. Let R be a commutative ring with identity and S be a multiplicative closed subset of R. A proper ideal I of R disjoint from S is called strongly quasi S-primary if there exists an s ∈ S such that whenever x , y ∈ R and xy ∈ I , then either s x 2 ∈ I or sy ∈ I . Many basic properties of strongly quasi S-primary ideals are given, and examples are presented to distinguish the last concept from other classical ideals. Moreover, forms of strongly quasi S-primary ideals in polynomial rings, power series rings and idealization of a module are investigated. [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring and a an ideal of R. The goal of this paper is to establish the local-global principle for the artinianness dimension r a (M) , where r a (M) is the smallest integer such that the local homology module of M is not artinian. For an artinian R-module M with the set Coass R H r a (M) a (M) finite, we show that r a (M) = inf { r a R p ( Hom R (R p , M)) | p ∈ Spec R } . And the class of all modules N such that Coass R N is finite is studied. [ABSTRACT FROM AUTHOR]

For any module M over a commutative ring R with identity, we consider Sch (M) as a certain class of prime submodules modulo the equivalence relation that two such prime submodules are the same if their colon ideals are equal. We topologize Sch (M) and introduce a sheaf O Sch (M) of commutative rings on it, which makes (Sch (M) , O Sch (M)) into a scheme. In particular, if M is a module over a Noetherian ring R, then (Sch (M) , O Sch (M)) is a locally Noetherian scheme. Among others, we give sufficient conditions for Sch (M) to be an affine scheme. [ABSTRACT FROM AUTHOR]

This paper establishes a connection between rings, lattices and common meadows. Meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. Common meadows are meadows that introduce, as the inverse of zero, an error term a which is absorbent for addition. We show that common meadows are unions of rings which are ordered by a partial order that defines a lattice. These results allow us to extend some classical algebraic constructions to the setting of common meadows. We also briefly consider common meadows from a categorical perspective. [ABSTRACT FROM AUTHOR]

This paper introduces a simple graph associated to a commutative ring. Nonzero ideals I and J of a commutative ring R are called Krull dimension-dependent whenever dim R / (I + J) = min { dim R / I , dim R / J } . Based on this, we introduce the dimension graph of R. We study and investigate this graph, and we determine all rings with disconnected dimension graph. Moreover, we introduce strongly Krull dimension-dependent ideals, and we examine these ideals using a subgraph of the dimension graph. [ABSTRACT FROM AUTHOR]

Consider a commutative monoid (M , + , 0) and a biadditive binary operation μ : M × M → M. We will show that under some additional general assumptions, the operation μ is automatically both associative and commutative. The main additional assumption is localizability of μ , which essentially means that a certain canonical order on M is compatible with adjoining some multiplicative inverses of elements of M. As an application we show that a division ring F is commutative provided that for all a ∈ F there exists a natural number k such that a − k is not a sum of products of squares. This generalizes the classical theorem that every archimedean ordered division ring is commutative to a more general class of formally real division rings that do not necessarily allow for an archimedean (total) order. Similar results about automatic associativity and commutativity are well-known for special types of partially ordered extended rings ("extended" in the sense that neither associativity nor commutativity of the multiplication is required by definition), namely in the uniformly bounded and the lattice-ordered cases, i.e. for (extended) f -rings. In these cases the commutative monoid in question is the positive cone of the partially ordered extended ring. We also discuss how these classical results can be obtained from our main theorem. [ABSTRACT FROM AUTHOR]

Given the n -dimensional affine space over an arbitrary commutative ring k , G a group scheme flat and finitely presented over k and X ⊂ A k n a G -invariant affine and closed subscheme, we prove that the GIT quotient ▪ is of finite type over k , even if k is not noetherian, provided G is linearly reductive. This is well-known if ▪ is faithfully flat, which does not hold in general. We also explore the infinite-dimensional case. Concretely, we consider a G − k finitely presented projective module M and an arbitrary k -module N. We prove, under certain conditions on k and G , that the degrees of the generators of (S • (M ∨ ⊗ N)) G and the degrees of the generators of the ideal of relations are bounded. We encode this property into the notion of partially generated graded (pgg) algebra and we give their main properties. In particular, we prove the existence of canonical equivariant immersions of spectra of pgg algebras in certain projective spaces. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, I an ideal of R and M a finitely generated R-module with dimR(M) = d. Denote by depthR(I,M) the depth of M in I. In [C. Huneke and V. Trivedi, The height of ideals and regular sequences, Manuscr. Math. 93 (1997) 137–142], Huneke and Trivedi proved that if R is a quotient of a regular ring then there exists a finite subset ΛM of Spec(R) such that depthR(I,M) =min픭∈Λ M{depthR픭(M픭) + ht((I + 픭)/픭)}. Denote by PsuppRi(M) = {픭 ∈Spec(R)|H픭 R픭i−dim(R/픭)(M픭)≠0} the ith pseudo support of M defined by Brodmann and Sharp [On the dimension and multiplicity of local cohomology modules, Nagoya Math. J. 167 (2002) 217–233]. In this paper, we prove that if PsuppRi(M) is closed for all i ≤ d then the above formula of depthR(I,M) holds true, where ΛM =⋃0≤i≤dmin PsuppRi(M). In particular, if R is a quotient of a Cohen–Macaulay local ring then ΛM = ⋃0≤i≤dminVar(AnnR(H픪i(M))). We also give some examples to clarify the results. [ABSTRACT FROM AUTHOR]

The main goal of this article is to introduce the concept of EM − G− graded rings. This concept is an extension of the notion of EM− rings. Let G be a group, and R be a G− graded commutative ring. The G− gradation of R can be extended to R[x] by taking the components (R[x])σ = Rσ[x]. We define R to be an EM − G− graded ring if every homogeneous zero divisor polynomial has an annihilating content. We provide examples of EM − G− graded rings that are not EM− rings, and we prove some interesting results regarding these rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and let V be a free R -module of rank n for some n ∈ N. Fixing an R -basis E of V , the symmetric group S n acts on V by permuting E and hence on tensor space V ⊗ r for r ∈ N via the usual tensor product action turning V and V ⊗ r into R S n -modules. For units q in R we construct an action of the corresponding Iwahori-Hecke algebra H R , q (S n) which specializes to the action of R S n , if q is taken to 1. The centralizing algebra of this action is called the q -partition algebra P R , q (n , r). Let R be a field of characteristic not dividing q. We prove, that P R , q (n , r) is isomorphic to the q -partition algebra defined by Halverson and Thiem by different means a few years ago. [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring, I an ideal of R , and M a finitely generated R -module. We consider the asymptotic injective dimensions, projective dimensions, Bass numbers, and Betti numbers of localizations of M / I n M at prime ideals of R and prove that these invariants are stable or have polynomial growth for large integers n that do not depend on the prime ideals. [ABSTRACT FROM AUTHOR]

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Suppose R is a finite commutative local ring, then it is known that R has four positive integers p , n , m , k called the invariants of R, where p is a prime number. This paper investigates the structure and classification up to isomorphism of local rings with residue field F p m and of length 4. Specifically, it gives a comprehensive characterization of Frobenius local rings of order p 4 m . Furthermore, we provide a detailed enumeration of the classes of all such rings with respect to their invariants p , n , m , k . Finite Frobenius rings are particularly advantageous for coding theory. This suitability arises from the fact that two classical theorems by MacWilliams, the Extension Theorem and the MacWilliams relations for symmetrized weight enumerators, can be generalized from finite fields to finite Frobenius rings. [ABSTRACT FROM AUTHOR]

We propose some general approach to studying the problem for the representability of every element in a field in the form , with and , where are fixed naturals, to imply the analogous representability of every square matrix over . As an application, we describe the fields and commutative rings with such that every square matrix over them is the sum of a -potent matrix and a -potent matrix for some small values of and . [ABSTRACT FROM AUTHOR]

The goal of this paper is to deepen the study of multiplicative lattices in the sense of Facchini, Finocchiaro and Janelidze. We provide a sort of Prime Ideal Principle that guarantees that maximal implies prime in a variety of cases (among them the case of commutative rings with identity). This result is used to study the lattice theoretic counterpart of multiplicative closed sets, that of m -systems. The notion of m -system is also studied from the topological point of view. [ABSTRACT FROM AUTHOR]

For a finite simple undirected graph G , the universal adjacency matrix U (G) is a linear combination of the adjacency matrix A (G) , the degree diagonal matrix D (G) , the identity matrix I and the all-ones matrix J , that is U (G) = α A (G) + β D (G) + γ I + η J , where α , β , γ , η ∈ ℝ and α ≠ 0. The cozero-divisor graph Γ ′ (R) of a finite commutative ring R with unity is a simple undirected graph with the set of all nonzero nonunits of R as vertices and two vertices x and y are adjacent if and only if x ∉ y R and y ∉ x R. In this paper, we study structural properties of Γ ′ (R) by defining an equivalence relation on its vertex set in terms of principal ideals of the ring R. Then we obtain the universal adjacency eigenpairs of Γ ′ (R) and its complement, and as a consequence one may obtain several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree diagonal matrix of Γ ′ (R) and Γ ′ (R) ¯ in an unified way. Moreover, we get the universal adjacency eigenpairs of the cozero-divisor graph and its complement for a reduced ring and the ring of integers modulo m in a simpler form. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with 1. A zero-divisor graph Λ (R) is an undirected graph on the vertex set R − { 0 } such that any two nonzero elements p , q ∈ R are adjacent whenever p ⋅ q = 0. The concept of entropy measure is fundamental and has been the subject of interest in various fields. The purpose of this paper is to calculate the entropy measure of Λ (R) for various rings. We take into consideration the ring ℤ h for h = φ ψ , φ 2 ψ , φ 2 ψ 2 , φ ψ χ , where φ , ψ and χ are distinct prime numbers. Moreover, we also discussed the entropy measure of the zero-divisor graph for the rings ℤ φ × ℤ ψ , ℤ φ × ℤ ψ × ℤ χ , ℤ φ ψ × ℤ χ , and ℤ φ 2 × ℤ ψ 2. This study will help us further understand the algebraic structure of the commutative rings. [ABSTRACT FROM AUTHOR]

In this work, we give three new techniques for constructing Hermitian self-dual codes over commutative Frobenius rings with a non-trivial involutory automorphism using λ -circulant matrices. The new constructions are derived as modifications of various well-known circulant constructions of self-dual codes. Applying these constructions together with the building-up construction, we construct many new best known quaternary Hermitian self-dual codes of lengths 26, 32, 36, 38 and 40. [ABSTRACT FROM AUTHOR]

Let A ⊆ B be commutative rings with unity, let S ⊆ T be torsion-free cancellative monoids, and let n ≥ 1 . We give a characterization of when the monoid ring A [ S ] is n-root closed in the monoid ring B [ T ] . For torsion-free cancellative ordered monoids (S , ≤) ⊆ (T , ≤) , we also present sufficient conditions and necessary conditions for the generalized power series ring [ ​ [ A S , ≤ ] ​ ] to be n-root closed in the generalized power series ring [ ​ [ B T , ≤ ] ​ ] . [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of functional prime ideals in a commutative ring. For a (left) R-module M and a functional ϕ (i.e., an R-linear map ϕ from M to R), an ideal I of R is said to be a ϕ -prime ideal if whenever a ∈ R and m ∈ M such that a ϕ (m) ∈ I , then a ∈ I or ϕ (m) ∈ I . This notion shows its ability to characterize different classes of ideals in terms of functional primeness with respect to specific R-modules. For instance, if the module M is the ideal I itself, then I is ϕ -prime for every ϕ ∈ Hom R (I , R) if and only if I is a trace ideal, and if the module M is the dual of I, then I is ϕ -prime for every ϕ ∈ Hom R (I − 1 , R) if and only if I is a prime ideal of R, or I is a strongly divisorial ideal. Several results are obtained and examples to illustrate the aims and scopes are provided. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, X be an indeterminate over R, R[X] be the polynomial ring over R, R[[X]] be the power series ring over R, and Q be a principal prime ideal of R[X] with (Q ∩ R)[X] ⊊ Q. It is well known that if R is an integral domain, then R[X]Q is a DVR and R[X] has infinitely many such principal prime ideals. In this paper, among other things, we show that (i) RQ∩R is a field, (ii) R[X]Q is a DVR, but (iii) there is a ring R such that R[X] has no principal prime ideal. We also study the maximal ideals of R[[X]] that are principal. [ABSTRACT FROM AUTHOR]

AbstractLocal log-regular rings are a certain class of Cohen-Macaulay local rings that are treated in logarithmic geometry. Our paper aims to provide purely commutative ring theoretic proof of several ring-theoretic properties of local log-regular rings such as an explicit description of a canonical module, and the finite generation of the divisor class group. [ABSTRACT FROM AUTHOR]

For each commutative ring R we associate a simple graph Γ1∗(R). We investigate the properties of an Artinian ring R when Γ1∗(R) is a star. [ABSTRACT FROM AUTHOR]

The paper studies the lattice of subgroups of an isotropic reductive group G (R) over a commutative ring R , normalized by the elementary subgroup E (R). We prove the sandwich classification theorem for this lattice under the assumptions that the isotropic rank of G is at least 2 and the structure constants are invertible in R. The theorem asserts that the lattice splits into a disjoint union of sublattices (sandwiches) E (R , q) ⩽ ... ⩽ C (R , q) parametrized by the ideals q of R , where E (R , q) denotes the relative elementary subgroup and C (R , q) is the inverse image of the center under the natural homomorphism G (R) → G (R / q). The main ingredients of the proof are the "level computation" by the first author and the generic element method developed by the second author. [ABSTRACT FROM AUTHOR]

We study structurable algebras and their associated Freudenthal triple systems over commutative rings. The automorphism groups of these triple systems are exceptional groups of type E 7 , and we realize groups of type E 6 as centralizers. When 6 is invertible, we further give a geometric description of homogeneous spaces of type E 7 / E 6 , and show that they parametrize principal isotopes of Brown algebras. As opposed to the situation over fields, we show that such isotopes may be non-isomorphic. [ABSTRACT FROM AUTHOR]

AbstractLet R be a commutative ring with identity and R′ be the integral closure of R. In this paper, we show that if R is an affine algebra over a field K, then every regular ideal of R′ is finitely generated, i.e., R′ is an r-Noetherian ring. We also study when the integral closure of an affine algebra is Noetherian. First we show that if R is a Krull ring such that R/P is a Noetherian domain for each minimal regular prime ideal P of R, then R is an r-Noetherian ring, which is a generalization of Nishimura’s result. As an application of this result, we prove that if R is an r-Noetherian ring with reg-dim R≤2 , then R′ is an r-Noetherian ring. We finally construct a couple of r-Noetherian rings, e.g., an r-Noetherian ring R that is not Noetherian and reg-dim R=∞ or reg-dim R=n≤ dim R=n+m−1 for arbitrary positive integers n, m. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with identity. In this paper, formal expressions of the number of m × n matrices over R of rank r and the number of invertible matrices over R are presented. The number of matrices over R with a given rank and a given number of single unit-entry rows, rows in which a single entry is a unit and all other entries are zero, is finally determined. [ABSTRACT FROM AUTHOR]

AbstractSuppose that K is a field, S=K[x1,…,xn] or S=K[[x1,…,xn]]. We present a characterization of those monomial ideals I of S, for which S/I is a half-factorial ring. This characterization is related to the structure of the base field K and also depends on the combinatorics of a graph constructed from the monomial ideal I. We also show that if R[x] or R[[x]] is half-factorial for an arbitrary commutative ring R, then R is an integral domain. [ABSTRACT FROM AUTHOR]

We show that the cotilting heart associated to a tilting complex T is a locally coherent and locally coperfect Grothendieck category (i.e., an Ind-completion of a small artinian abelian category) if and only if T is product-complete. We then apply this to the specific setting of the derived category of a commutative noetherian ring R. If dim(R) < ∞, we show that there is a derived duality Dfb​g (R) ≅ Db(B)op between modR and a noetherian abelian category B if and only if R is a homomorphic image of a Cohen–Macaulay ring. Along the way, we obtain new insights about t-structures in Dfb​g​(R). In the final part, we apply our results to obtain a new characterization of the class of those finite-dimensional noetherian rings that admit a Gorenstein complex.Cite this article [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring. Denote by mod R the category of finitely generated R-modules. In the present paper, we first provide various sufficient (and necessary) conditions for a full subcategory of mod R to be a Serre subcategory, which include several refinements of theorems of Stanley and Wang and of Takahashi with simpler proofs. Next we consider when an IKE-closed subcategory of mod R is a torsion-free class. We investigate certain modules out of which all modules of finite length can be built by taking direct summands and extensions, and then we apply it to show that the IKE-closed subcategories of mod R are torsion-free classes in the case where R is a certain numerical semigroup ring. [ABSTRACT FROM AUTHOR]

Let G be a group with identity e. Let ℜ be a G-graded commutative ring, ℑ be a graded ℜ-module. In this paper, we introduce and study the concept of graded n-submodules of ℑ. We obtain many results concerning graded n-submodules. Some characterizations of graded n-submodules and their homogeneous components are given. A proper graded submodule U of ℑ is said to be a graded n-submodule if whenever r ∈ h(ℜ), m ∈ h(ℑ) with rm ∈ U and r /∈ Gr(Annℜ(ℑ)), then m ∈ U. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring, I an ideal of R , and M a finitely generated R -module. We consider the idealization R ⋉ M of M over R. The goal of this paper is to investigate algebraic properties of R ⋉ M that are related to those of R and M. Specifically, we provide characterizations for the Cohen–Macaulayness, sequentially Cohen–Macaulayness, generalized Cohen–Macaulayness, and graded maximal depth property of R ⋉ M with respect to I ⋉ M , in terms of the corresponding properties for R and M with respect to I. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, a proper ideal of R and N a nonzero finitely generated R -module with N ≠ N. Let d (respectively, c) be the smallest (respectively, greatest) non-negative integer i such that the local cohomology H i (N) is nonzero. In this paper, we provide sharp bounds under inclusion for the annihilators of the local cohomology modules H d (N) , H c (N) and these annihilators are computed in certain cases. Also, we construct a counterexample to Lynch's conjecture. [ABSTRACT FROM AUTHOR]

In this article, the notion of γ -dual codes over finite chain rings is introduced as an extension of dual codes over finite chain rings. Various characteristics and properties of γ -dual codes over finite chain rings are explored. We provide both necessary and sufficient conditions for the existence of γ -self-dual codes over finite chain rings. Additionally, we investigate the γ -dual of skew ϕ - α -constacyclic codes over finite chain rings. [ABSTRACT FROM AUTHOR]

We investigate the Möbius groups theory in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero-divisors. In this paper, we first generalize classical conjugacy classification to bicomplex analysis. Then we have a discussion of the iterates of a bicomplex Möbius transformation and study the attractive and repulsive fixed points in bicomplex setting. Finally, we shall prove two useful results concerning fixed point. [ABSTRACT FROM AUTHOR]

We describe the prime ideals and, in particular, the maximal ideals in products R = ∏ D λ of families (D λ) λ ∈ Λ of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra ∏ (max (D λ)) , where max (D λ) is the spectrum of maximal ideals of D λ , and denotes the power set. If every D λ is in a certain class of rings including finite character domains and one-dimensional domains, we completely characterize the maximal ideals of R. If every D λ is a Prüfer domain, we completely characterize all prime ideals of R. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and S a multiplicative subset of R. The aim of this paper is to study the class of commutative rings in which every S-flat module is flat (resp., projective). An R-module M is said to be S-flat if the localization of M at S, MS, is a flat RS-module. Commutative rings R for which all S-flat R-modules are flat are characterized by the fact that R/Rs is a von Neumann regular ring for every s ∈ S . While, commutative rings R for which all S-flat R-modules are projective are characterized by the following two conditions: R is perfect and the Jacobson radical J(R) of R is S-divisible. Rings satisfying these conditions are called S-perfect. Moreover, we give some examples to distinguish perfect rings, S-perfect rings, and semisimple rings. We also investigate the transfer results of the "S-perfectness" for various ring constructions, which allows the construction of more interesting examples. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, and A , B be unital algebras over R . Let M be a unital (A , B) -bimodule, which is faithful as a left A -module and also as a right B -module. Let T = Tri (A , M , B) be an (n − 1) -torsion free triangular algebra. In this note, we investigate the structure of local Lie n-derivations under certain mild conditions and 2-local Lie n-derivations under weaker conditions than that of local Lie n-derivations with n ≥ 3 on T . Under their corresponding conditions, we conclude that they are both standard on T . [ABSTRACT FROM AUTHOR]

Let (X , ≤) be a locally finite pre-ordered set and R be a commutative ring with unity. In this paper we apply the theory of zero product determined algebras to show that each linear map on the incidence algebra I (X , R) which is derivable at zero is a generalized derivation and every local derivation on I (X , R) is a derivation. [ABSTRACT FROM AUTHOR]

Let a be an ideal of a given commutative Noetherian ring R which satisfies the condition of the first problem of R. Hartshorne in [Affine duality and cofiniteness, Invent. Math. 9 (1970), 145–164]. In this paper, we prove that a also satisfies the condition of his second problem in the same article. We also provide an example to show that the converse statement does not hold in general. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the zero-divisor associate graph ΓD(R) over a finite commutative ring R. It is a simple undirected graph whose vertex set consists of all non-zero elements of R, and two vertices a, b are adjacent if and only if there exist non-zero zero-divisors z1, z2 in R such that az1 = bz2. We determine the necessary and sufficient conditions for connectedness and completeness of ΓD(R) for a unitary commutative ring R. The chromatic number of ΓD(R) is also studied. Next, we characterize the rings R for which ΓD(R) becomes a line graph of some graph. Finally, we give the complete list of graphs with at most 15 vertices which are realizable as ΓD(R), characterizing the associated ring R in each case. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph with diameter d, and k ∈ [1, d] be an integer. A radio k-coloring of graph G is a mapping g : V(G) → {0} ∪ N satisfying |g(u) - g(v)| ≥ 1 + k - d(u, v) for any pair of distinct vertices u and v of the graph G, where d(u, v) denotes distance between vertices u and v in G. The number max{g(u) : u ∈ V(G)} is known as the span of g and is denoted by rck(g). The radio k-chromatic number of graph G, denoted by rck(G), is defined as min{rck(g) : g is a radio k-coloring of G}. For k = d-1, the radio k-coloring of graph G is called an antipodal coloring. So rcd-1(G) is called the antipodal number of G and is denoted by ac(G). Here, we study antipodal coloring of the Cartesian product of the complete graph Kr and cycle Cs, Kr...Cs, for r ≥ 4 and s ≥ 3. We determine the antipodal number of Kr...Cs, for even r ≥ 4 with s ≡ 1 (mod 4); and for any r ≥ 4 with s = 4t + 2, t odd. Also, for the remaining values of r and s, we give lower and upper bounds for ac(Kr...Cs). [ABSTRACT FROM AUTHOR]

The folded hypercube FQn is the Cayley graph Cay(Z2n, S), where S = {e1, e2, ..., en} ∪ {u = e1 + e2 + ... + en}, and ei = (0, ..., 0, 1, 0, ..., 0), with 1 at the ith position, 1 ≤ i ≤ n. In this paper, we show that the folded hypercube FQn is a distance-transitive graph. Then, we study some properties of this graph. In particular, we show that if n ≥ 4 is an even integer, then the folded hypercube FQn is an automorphic graph, that is, FQn is a distance-transitive primitive graph which is not a complete or a line graph. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with or without unity and Γe(R) be its extended zero-divisor graph with vertex set Z*(R) = Z(R)\{0} and two distinct vertices x, y are adjacent if and only if x.y = 0 or x + y ∈ Z*(R). In this paper, we characterize finite commutative rings whose extended zero-divisor graph have clique number 1 or 2. We completely characterize the rings of the form R ≅ R1 x R2, where R1 and R2 are local, having clique number 3, 4 or 5. Further we determine the rings of the form R ≅ R1 x R2 x R3, where R1, R2 and R3 are local rings, to have clique number equal to six. [ABSTRACT FROM AUTHOR]