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The rings considered in this paper are finite commutative rings with identity, which are not fields. For any ring [Formula: see text] which is not a field and which is not necessarily finite, we denote the set of all zero-divisors of [Formula: see text] by [Formula: see text] and [Formula: see text] by [Formula: see text]. Let [Formula: see text] denote the zero-divisor graph of [Formula: see text] and for a finite ring [Formula: see text], let [Formula: see text] denote the maximum degree of [Formula: see text]. We denote [Formula: see text] by [Formula: see text]. The aim of this paper is to study some properties of [Formula: see text].

The paper deals with rank, trace, eigenvalues and norms of the matrix [Formula: see text], where [Formula: see text] are ith components of any real sequence [Formula: see text]. A result in this paper is that the Euclidean and spectral norms of the matrix [Formula: see text] is [Formula: see text]. This is a generalization of the main result by Solak [Appl. Math. Comput. 232 (2014) 919–921], with the proof based on a simple property of norms of real matrices.

Suppose [Formula: see text] is a finite field extension of [Formula: see text] containing a primitive [Formula: see text]th root of unity. Let [Formula: see text] be a maximal [Formula: see text]-extension of [Formula: see text] with the Galois group of period [Formula: see text] and nilpotent class [Formula: see text]. In this paper, we develop formalism which allows us to study the structure of [Formula: see text] via methods of Lie theory. In particular, we introduce an explicit construction of a Lie [Formula: see text]-algebra [Formula: see text] and an identification [Formula: see text], where [Formula: see text] is a [Formula: see text]-group obtained from the elements of [Formula: see text] via the Campbell–Hausdorff composition law. In the next paper, we apply this formalism to describe the ramification filtration [Formula: see text] and an explicit form of the Demushkin relation for [Formula: see text].

WOS: 000389311800021, In [Finite presentability of Bruck-Reilly extensions of groups, J. Algebra 242 (2001) 2030], Araujo and RuAtic studied finite generation and finite presentability of Bruck Reilly extension of a group. In this paper, we aim to generalize some results given in that paper to generalized Bruck Reilly s -extension of a group. In this way, we determine necessary and sufficent conditions for generalized Bruck Reilly s-extension of a group, G R. (G; beta, gamma; u), to be finitely generated and finitely presented. Let G be a group, (3,-y: be morphisms and u is an element of H-1 (H-1(*) and H-1 are the ?-G- and ?-t-classes, respectively, contains the identity element kr of T). We prove that GE R* (G; 3, 7; u) is finitely generated if and only if there exists a finite subset Xo C G such that G is generated by ((boolean OR(i >= 0) X-0,beta(i))U(Uj >= 0X0 gamma(j)). We also prove that C superset of R.* (C; beta, gamma; u) is finitely presented if and only if G is presented by (X; R), where X is a finite set and R = (U i >= 0 R-0 beta(i)) boolean OR (U-j >= 0 R0 gamma(j)) = {w(1)beta(i) = upsilon(1)beta(i) : i >= 0, w(1) = v(1) is an element of H-1* for some finite set of relations R0 subset of X* x X*.

Denote by $G$ a finite group and by $\psi(G)$ the sum of element orders in $G$. If $t$ is a positive integer, denote by $C_t$ the cyclic group of order $t$ and write $\psi(t)=\psi(C_t)$. In this paper we proved the following Theorem A: Let $G$ be a non-cyclic group of odd order $n=qm$, where $q$ is the smallest prime divisor of $n$ and $(m,q)=1$. Then the following statements hold. (1) If $q=3$, then $\frac {\psi(G)}{\psi(|G|)}\leq \frac {85}{301}$, and equality holds if and only if $n=3\cdot 7\cdot m_1$ with $(m_1,42)=1$ and $G=(C_7\rtimes C_3)\times C_{m_1}$, with $C_7\rtimes C_3$ non-abelian. (2) If $q>3$, then $\frac {\psi(G)}{\psi(|G|)}\leq \frac {p^4+p^3-p^2+1}{p^5+1}$, where $p$ is the smallest prime bigger than $q$ and equality holds if and only if $n=qp^2m_1$ with $(m_1,p!)=1$ and $G=C_q\times C_p\times C_p \times C_{m_1}$.

International audience; Two finite words [Formula: see text] and [Formula: see text] are [Formula: see text]-binomially equivalent if, for each word [Formula: see text] of length at most [Formula: see text], [Formula: see text] appears the same number of times as a subsequence (i.e., as a scattered subword) of both [Formula: see text] and [Formula: see text]. This notion generalizes abelian equivalence. In this paper, we study the equivalence classes induced by the [Formula: see text]-binomial equivalence. We provide an algorithm generating the [Formula: see text]-binomial equivalence class of a word. For [Formula: see text] and alphabet of [Formula: see text] or more symbols, the language made of lexicographically least elements of every [Formula: see text]-binomial equivalence class and the language of singletons, i.e., the words whose [Formula: see text]-binomial equivalence class is restricted to a single element, are shown to be non-context-free. As a consequence of our discussions, we also prove that the submonoid generated by the generators of the free nil-[Formula: see text] group (also called free nilpotent group of class [Formula: see text]) on [Formula: see text] generators is isomorphic to the quotient of the free monoid [Formula: see text] by the [Formula: see text]-binomial equivalence.

Let [Formula: see text] be a finite group and [Formula: see text] be the set of the elements [Formula: see text] of [Formula: see text] such that [Formula: see text] where [Formula: see text]. In this paper, we give strong restrictions on the structure of the quotients [Formula: see text] where [Formula: see text] is a finite group with an automorphism [Formula: see text] such that [Formula: see text] and [Formula: see text] is the largest normal [Formula: see text]-subgroup of [Formula: see text].

We call a reduced word [Formula: see text] multiplicity-bounding if and only if a finite group on which the word map of [Formula: see text] has a fiber of positive proportion [Formula: see text] can only contain each non-abelian finite simple group [Formula: see text] as a composition factor with multiplicity bounded in terms of [Formula: see text] and [Formula: see text]. In this paper, based on recent work of Nikolov, we present methods to show that a given reduced word is multiplicity-bounding and apply them to give some nontrivial examples of multiplicity-bounding words, such as words of the form [Formula: see text], where [Formula: see text] is a single variable and [Formula: see text] an odd integer.

This paper deals with the reducibility property of semidirect products of the form [Formula: see text] relatively to graph equation systems, where D denotes the pseudovariety of definite semigroups. We show that if the pseudovariety [Formula: see text] is reducible with respect to the canonical signature [Formula: see text] consisting of the multiplication and the [Formula: see text]-power, then [Formula: see text] is also reducible with respect to [Formula: see text].

Let [Formula: see text] be a field of characteristic 0 and let [Formula: see text]. The algebra [Formula: see text] admits a natural grading [Formula: see text] by the cyclic group [Formula: see text] of order 2. In this paper, we describe the [Formula: see text]-graded A-identities for [Formula: see text]. Recall that an A-identity for an algebra is a multilinear polynomial identity for that algebra which is a linear combination of the monomials [Formula: see text] where [Formula: see text] runs over all even permutations of [Formula: see text] that is [Formula: see text], the [Formula: see text]th alternating group. We first introduce the notion of an A-identity in the case of graded polynomials, then we describe the graded A-identities for [Formula: see text], and finally we compute the corresponding graded A-codimensions.

In this paper, we study quadratic residue (QR) codes of prime length [Formula: see text] over the ring [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are distinct odd prime numbers. We analyze some basic properties of cyclic codes of length [Formula: see text] over [Formula: see text], we define QR codes by their generating idempotents. Further, we discuss the extended QR codes. We present a considerable number of good [Formula: see text]-ary codes as Gray images of QR codes over [Formula: see text] by considering the case when [Formula: see text] is an odd prime.

The forgotten topological index of a graph [Formula: see text], denoted by [Formula: see text], is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the degree of the vertex [Formula: see text]. In this paper, we establish upper bounds on the forgotten topological index of trees, unicyclic graphs and bicyclic graphs with a given domination number.

Let [Formula: see text] be a [Formula: see text], [Formula: see text]- graph, [Formula: see text] the detour diameter of [Formula: see text], and cdn[Formula: see text] is the connected detour number of [Formula: see text]. Santhakumaran and Athisayanathan are characterized by the graphs of [Formula: see text] 4 and cdn[Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we obtain a necessary and sufficient condition for a graph [Formula: see text], [Formula: see text] 2, of [Formula: see text] 8 to have cdn[Formula: see text]. Moreover, graphs G of [Formula: see text] 6, and [Formula: see text] = 5 and 6, such that cdn[Formula: see text], are characterized in this research.

Let [Formula: see text] be a finite group and [Formula: see text] a subgroup of [Formula: see text]. Then [Formula: see text] is said to be [Formula: see text]-permutably embedded in [Formula: see text] if a Sylow [Formula: see text]-subgroup of [Formula: see text] is also a Sylow [Formula: see text]-subgroup of some [Formula: see text]-permutable subgroup of [Formula: see text] for every prime dividing the order of [Formula: see text]. We say that [Formula: see text] is nearly[Formula: see text]-embedded in [Formula: see text] if [Formula: see text] has a normal subgroup [Formula: see text] such that [Formula: see text] is [Formula: see text]-permutable in [Formula: see text] and [Formula: see text], where [Formula: see text] is the subgroup of [Formula: see text] generated by all those subgroups of [Formula: see text] which are [Formula: see text]-permutably embedded in [Formula: see text]. In this paper, we study the properties of the nearly [Formula: see text]-embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized.

Let [Formula: see text] be a commutative ring and [Formula: see text] be an [Formula: see text]-module. The compressed essential graph of [Formula: see text], denoted by [Formula: see text] is a simple undirected graph associated to [Formula: see text] whose vertices are classes of torsion elements of [Formula: see text] and two distinct classes [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an essential ideal of [Formula: see text]. In this paper, we study diameter and girth of [Formula: see text] and we characterize all modules for which the compressed essential graph is connected. Moreover, it is proved that [Formula: see text], whenever [Formula: see text] is Noetherian and [Formula: see text] is a finitely generated multiplication module with [Formula: see text].

Let [Formula: see text] be the moduli space of smooth, genus [Formula: see text] curves over an algebraically closed field [Formula: see text] of zero characteristic. Denote by [Formula: see text] the subset of [Formula: see text] of curves [Formula: see text] such that [Formula: see text] (as a finite nontrivial group) is isomorphic to a subgroup of [Formula: see text] and let [Formula: see text] be the subset of curves [Formula: see text] such that [Formula: see text], where [Formula: see text] is the full automorphism group of [Formula: see text]. Now, for an integer [Formula: see text], let [Formula: see text] be the subset of [Formula: see text] representing smooth, genus [Formula: see text] curves that admit a non-singular plane model of degree [Formula: see text] (in this case, [Formula: see text]) and consider the sets [Formula: see text] and [Formula: see text]. In this paper we first determine, for an arbitrary but a fixed degree [Formula: see text], an algorithm to list the possible values [Formula: see text] for which [Formula: see text] is non-empty, where [Formula: see text] denotes the cyclic group of order [Formula: see text]. In particular, we prove that [Formula: see text] should divide one of the integers: [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. Secondly, consider a curve [Formula: see text] with [Formula: see text] such that [Formula: see text] has an element of “very large” order, in the sense that this element is of order [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. Then we investigate the groups [Formula: see text] for which [Formula: see text] and also we determine the locus [Formula: see text] in these situations. Moreover, we work with the same question when [Formula: see text] has an element of “large” order: [Formula: see text], [Formula: see text] or [Formula: see text] with [Formula: see text] an integer.

Let [Formula: see text] be a smooth hypersurface of degree [Formula: see text] in a projective space [Formula: see text] and take a point [Formula: see text] in [Formula: see text]. Let [Formula: see text] be the relative Hilbert scheme parametrizing zero-dimensional subscheme, of length [Formula: see text], of fibers of the projection morphism [Formula: see text] from [Formula: see text]. In this paper we present an embedding of the relative Hilbert scheme [Formula: see text] into a weighted projective space and describe its defining ideal for general [Formula: see text]. We also study line bundles on the relative Hilbert scheme [Formula: see text] for [Formula: see text] and general [Formula: see text].

For a given nonabelian finite group [Formula: see text] and [Formula: see text] where [Formula: see text] denotes the center of [Formula: see text] we introduce a new graph [Formula: see text] associated to the group [Formula: see text] as follows: Take [Formula: see text] as its vertex set and two distinct vertices [Formula: see text] and [Formula: see text] being adjacent if and only if there exists an element [Formula: see text] such that [Formula: see text] This paper is devoted to investigate the properties of graphs [Formula: see text] and establish some graph theoretical properties. Moreover, we describe the planarity of these graphs when [Formula: see text] Also, we provide some examples of finite nonabelian groups [Formula: see text] with the property that if [Formula: see text] and [Formula: see text] for some group [Formula: see text] and [Formula: see text] then [Formula: see text]

We define [Formula: see text] and [Formula: see text] for [Formula: see text], where [Formula: see text]. The labeling [Formula: see text] is called a vertex irregular reflexive [Formula: see text]-labeling if any vertices have distinct weight which [Formula: see text] for any vertices in a graph [Formula: see text]. The weight of a vertex [Formula: see text] is defined as the sum of the labels of vertex and the labels of all edges incident this vertex. The smallest [Formula: see text] for which such labeling exists is called reflexive vertex strength of [Formula: see text], denoted by [Formula: see text]. In this paper, we determine the reflexive vertex strength of gear graphs, book graphs, triangular book graph and the disjoint union of gear graphs.

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text]. The general reduced second Zagreb index of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is any real number and [Formula: see text] is the degree of the vertex [Formula: see text] of [Formula: see text]. In this paper, the general reduced second Zagreb index of the Cartesian product, corona product, join of graphs and two new operations of graphs are computed.

Let us consider a mapping [Formula: see text] of a graph [Formula: see text], where [Formula: see text] is an integer, [Formula: see text]. The mapping [Formula: see text] induces for every vertex [Formula: see text] of [Formula: see text] the label [Formula: see text]. Let [Formula: see text] ([Formula: see text]) denote the number of edges (vertices) in [Formula: see text] that are labeled with the number [Formula: see text] under the labeling [Formula: see text], [Formula: see text]. The function [Formula: see text] is called a [Formula: see text]-total edge product cordial labeling of [Formula: see text] if [Formula: see text] for [Formula: see text]. A graph [Formula: see text] with a [Formula: see text]-total edge product cordial labeling is called a [Formula: see text]-total edge product cordial graph. In this paper, we prove that the grid graph [Formula: see text] for [Formula: see text] admits a [Formula: see text]-total edge product cordial labeling.

Let [Formula: see text] denote the power graph of a finite group [Formula: see text]. Let [Formula: see text] denote the Signless Laplacian spectral radius of [Formula: see text]. In this paper, we give lower and upper bounds on [Formula: see text] for any [Formula: see text] and find those graphs for which the extremal values are attained. We give a comparison between the bounds obtained and exact value of [Formula: see text] for any [Formula: see text]. We then find the eigenvalues of [Formula: see text] and give lower and upper bounds on the spectral radius of [Formula: see text]. When [Formula: see text] and [Formula: see text] where [Formula: see text] are primes and [Formula: see text] is a positive integer, we obtain sharper bounds on [Formula: see text]. Finally, we make a conjecture on [Formula: see text] for any [Formula: see text].

The first multiplicative Zagreb index [Formula: see text] of a graph [Formula: see text] is the product of the square of every vertex degree, while the second multiplicative Zagreb index [Formula: see text] is the product of the products of degrees of pairs of adjacent vertices. In this paper, we give sharp lower bound for [Formula: see text] and upper bound for [Formula: see text] of trees with given distance [Formula: see text]-domination number, and characterize those trees attaining the bounds.

Let the self-affine measure [Formula: see text] be generated by an expanding matrix [Formula: see text] and a finite integer digit set [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] for an integer [Formula: see text], then [Formula: see text] admits an infinite orthogonal set of exponential functions if and only if there exists [Formula: see text] such that [Formula: see text] for some [Formula: see text] with [Formula: see text] and [Formula: see text].

Let [Formula: see text] be a finite field with characteristic [Formula: see text] having [Formula: see text] elements and [Formula: see text] be the dihedral group of order [Formula: see text]. In this paper, we have obtained the structure of unit groups of group algebra [Formula: see text], for any prime [Formula: see text].

Let [Formula: see text] be a numerical semigroup with embedding dimension [Formula: see text], minimal set of generators [Formula: see text], Frobenius number [Formula: see text], multiplicity [Formula: see text] and genus [Formula: see text]. In this paper, we prove that Wilfs conjecture i.e. the inequality [Formula: see text] holds for [Formula: see text] when [Formula: see text] is a basis for [Formula: see text]

Let [Formula: see text] be the family of analytic and normalized functions [Formula: see text] in the open unit disc [Formula: see text]. In this paper, we consider the following classes: [Formula: see text] and [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text]. We show that if [Formula: see text], then [Formula: see text] and [Formula: see text] are greater than [Formula: see text], and if [Formula: see text], then [Formula: see text]. Also, some another interesting properties of the class [Formula: see text] are investigated. Finally, the radius of univalence of 2nd section sum of [Formula: see text] is obtained.

A set [Formula: see text] is said to be a [Formula: see text]-hop dominating set ([Formula: see text]-HDS) of a graph [Formula: see text] if every vertex [Formula: see text] is within [Formula: see text]-distances from at least one vertex [Formula: see text], i.e. [Formula: see text], where [Formula: see text] is a fixed positive integer. A dominating set [Formula: see text] is said to be minimum [Formula: see text]-hop connected dominating set of a graph [Formula: see text], if it is minimal as well as it is [Formula: see text]-HDS and the subgraph of G made by [Formula: see text] is connected. In this paper, we present an [Formula: see text]-time algorithm for computing a minimum [Formula: see text]-hop connected dominating set of permutation graphs with [Formula: see text] vertices.

If [Formula: see text], [Formula: see text] is a sequence of mappings on a metric space [Formula: see text], a point [Formula: see text] in [Formula: see text] is called an inverse fixed point of the sequence [Formula: see text], if [Formula: see text] tends to zero as [Formula: see text] tends to infinity. This definition is proposed and studied in this paper to obtain a few fixed point results.

Let [Formula: see text] be a simple connected graph and [Formula: see text] a coloring of the vertices in [Formula: see text] For any [Formula: see text], let [Formula: see text] be the sum of colors of the vertices adjacent to [Formula: see text]. Then [Formula: see text] is called a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text]. The minimum number of colors needed in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text], denoted by [Formula: see text] In this paper, we prescribe a sigma coloring of the join of paths and cycles. As a consequence, we determine the sigma chromatic number of the join of a finite number of paths and cycles. In particular, let [Formula: see text], where [Formula: see text] or [Formula: see text] with [Formula: see text] If [Formula: see text], where [Formula: see text] and [Formula: see text], then [Formula: see text] if [Formula: see text] is an odd cycle, for some [Formula: see text] and [Formula: see text] otherwise.

A 2-rainbow edge dominating function (2REDF) of a graph [Formula: see text] is a function [Formula: see text] from the edge set [Formula: see text] to the set of all subsets of the set [Formula: see text] such that for any edge [Formula: see text] with [Formula: see text] the condition [Formula: see text] is fulfilled, where [Formula: see text] is the open neighborhood of [Formula: see text]. The weight of a 2REDF [Formula: see text] is the value [Formula: see text]. The minimum weight of a 2REDF is the 2-rainbow edge domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we initiate the study of 2-rainbow edge domination in graphs. We present various sharp bounds, exact values and characterizations for the 2-rainbow edge domination number of a graph.

Sanskruti index of a graph [Formula: see text] with vertex-set [Formula: see text] and edge-set [Formula: see text] is [Formula: see text], where [Formula: see text]. In this paper, the extremal graphs in the class of unicyclic graphs are characterized with respect to the Sanskruti index, where the considered class of unicyclic graphs contains five different large families of graphs.

In this paper, we study skew cyclic and skew constacyclic codes over the ring [Formula: see text], where [Formula: see text] and [Formula: see text], for [Formula: see text] and [Formula: see text]. We have introduced some new Gray maps and determined the relation between codes using the [Formula: see text]-Gray images of skew constacyclic codes.

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type [Formula: see text]. We say that a graph [Formula: see text] satisfies a term equation [Formula: see text] if the corresponding graph algebra [Formula: see text] satisfies [Formula: see text]. The set of all term equations [Formula: see text], which the graph [Formula: see text] satisfies, is denoted by [Formula: see text]. The class of all graph algebras satisfy all term equations in [Formula: see text] is called the graph variety generated by [Formula: see text] denoted by [Formula: see text]. A term is called a linear term if each variable which occurs in the term, occurs only once. A term equation [Formula: see text] is called a linear term equation if [Formula: see text] and [Formula: see text] are linear terms. This paper is devoted to a thorough investigation of graph varieties defined by linear term equations. In particular, we give a complete description of rooted graphs generating a graph variety described by linear term equations.

Let [Formula: see text] be a graph and let [Formula: see text] be a function. A vertex [Formula: see text] is protected with respect to [Formula: see text], if [Formula: see text] or [Formula: see text] and [Formula: see text] is adjacent to a vertex of positive weight. The function [Formula: see text] is a co-Roman dominating function, abbreviated CRDF if: (i) every vertex in [Formula: see text] is protected, and (ii) each [Formula: see text] with positive weight has a neighbor [Formula: see text] with [Formula: see text] such that the function [Formula: see text], defined by [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text], has no unprotected vertex. The weight of [Formula: see text] is [Formula: see text]. The co-Roman domination number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum weight of a co-Roman dominating function on [Formula: see text]. In this paper, we present some new sharp bounds on [Formula: see text]. Some of our results improve the previous bounds.

For positive integer [Formula: see text], a graph [Formula: see text] is said to be [Formula: see text]-magic if the edges of [Formula: see text] can be labeled with the nonzero elements of Abelian group [Formula: see text], where [Formula: see text] (the set of integers) and [Formula: see text] is the group of integers mod [Formula: see text], so that the sum of the labels of the edges incident to any vertex of [Formula: see text] is the same. When this constant sum is [Formula: see text], we say that [Formula: see text] is a zero-sum [Formula: see text]-magic graph. The set of all [Formula: see text] for which [Formula: see text] is a zero-sum [Formula: see text]-magic graph is the null set of [Formula: see text]. In this paper, we will completely determine the null set of the join of a finite number of paths.

In this paper, we establish the structure of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] where [Formula: see text] is a finite field and [Formula: see text] is the dihedral group of order [Formula: see text]. For an odd prime [Formula: see text], we also establish the structure of [Formula: see text], when characteristic of [Formula: see text] is [Formula: see text] or [Formula: see text].

Let [Formula: see text] and [Formula: see text] be two central relations on a finite set [Formula: see text]. It is known from Rosenberg’s classification theorem (1965) that the clones [Formula: see text] and [Formula: see text] which consist of all operations on [Formula: see text] that preserve [Formula: see text], respectively, [Formula: see text] are among the maximal clones on [Formula: see text]. In this paper, we find all central relations [Formula: see text] such that the clone [Formula: see text] is a maximal subclone of [Formula: see text], where [Formula: see text] is a fixed central relation.

Let [Formula: see text] be the complement of the zero-divisor graph of a finite commutative ring [Formula: see text]. In this paper, we provide the answer of the question (ii) raised by Osba and Alkam in [11] and prove that [Formula: see text] is a divisor graph if [Formula: see text] is a local ring. It is shown that when [Formula: see text] is a product of two local rings, then [Formula: see text] is a divisor graph if one of them is an integral domain. Further, if [Formula: see text], then [Formula: see text] is a divisor graph.

Let [Formula: see text] be a semigroup of all transformations on a set [Formula: see text] Let [Formula: see text] be a fixed nonempty subset of [Formula: see text] and [Formula: see text] Then [Formula: see text] is a subsemigroup of [Formula: see text] which leaves a subset [Formula: see text] of [Formula: see text] invariant. If [Formula: see text] then [Formula: see text] In this paper, we give necessary and sufficient conditions for elements in [Formula: see text] to be left or right magnifying.

Let [Formula: see text] be a connected finite graph and [Formula: see text] be the set of functions defined on [Formula: see text]. Let [Formula: see text] be the discrete [Formula: see text]-Laplacian on [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] is positive everywhere. Consider the operator [Formula: see text]. We prove that [Formula: see text] is one-to-one, onto and preserves order. So it implies that there exists a unique solution to the equation [Formula: see text] for any given [Formula: see text]. We also prove that the equation [Formula: see text] has a solution which is unique up to a constant, where [Formula: see text] is the average of [Formula: see text]. With the help of these results, we finally give various conditions such that the [Formula: see text]th Kazdan–Warner equation [Formula: see text] has a solution on [Formula: see text] for given [Formula: see text] and [Formula: see text]. Thus we generalize Grigor’yan, Lin and Yang’s work Kazdan–Warner equation on graph, Calc. Var. Partial Differential Equations 55(4) (2016) Paper No. 92, 13 pp. for [Formula: see text] to any [Formula: see text].

Let [Formula: see text] be the [Formula: see text]th Fibonacci number and [Formula: see text] be the [Formula: see text]th Lucas number. The order of appearance [Formula: see text] of a natural number [Formula: see text] is defined as the smallest natural number [Formula: see text] such that [Formula: see text] divides [Formula: see text]. For instance, [Formula: see text], for all [Formula: see text]. In this paper, among other things, we prove that [Formula: see text] depends on [Formula: see text], where [Formula: see text] is the greatest common divisor of numbers [Formula: see text] and [Formula: see text], which fulfill the condition [Formula: see text].

Let [Formula: see text] be a simple connected graph with [Formula: see text] vertices, [Formula: see text] edges and let [Formula: see text] be its Laplacian eigenvalues. The Laplacian resolvent energy of graph [Formula: see text] is defined by [Formula: see text]. In this paper, we give some new lower bounds for the invariant [Formula: see text].

Let [Formula: see text] and [Formula: see text] be integers such that [Formula: see text] [Formula: see text] and let [Formula: see text] be generalized Hecke group associated to [Formula: see text] and [Formula: see text] Generalized Hecke group [Formula: see text] is generated by [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] In this paper, for positive integer [Formula: see text] we study the power subgroups [Formula: see text] of generalized Hecke groups [Formula: see text]. Also, we give some results about free normal subgroups of generalized Hecke groups [Formula: see text]

In this paper, we discuss the measure theoretic composition operators on [Formula: see text] in some operator classes, such as class [Formula: see text]-[Formula: see text], class [Formula: see text]-[Formula: see text] and class [Formula: see text]-[Formula: see text]. We also study some spectral properties of class [Formula: see text]-[Formula: see text] composition operators on [Formula: see text].

Let [Formula: see text] be a finite simple graph. The [Formula: see text]-distance graph [Formula: see text] of [Formula: see text] is the graph with the same vertex set as [Formula: see text] and two vertices are adjacent if and only if their distance in [Formula: see text] is exactly [Formula: see text]. A graph [Formula: see text] is a [Formula: see text]-distance graph if there exists a graph [Formula: see text] such that [Formula: see text]. In this paper, we give three characterizations of [Formula: see text]-distance graphs, and find all graphs [Formula: see text] such that [Formula: see text] or [Formula: see text], where [Formula: see text] is an integer, [Formula: see text] is the path of order [Formula: see text], and [Formula: see text] is the complete graph of order [Formula: see text].

Let [Formula: see text] be a graph and [Formula: see text] A dominating set [Formula: see text] is a set of vertices such that each vertex of [Formula: see text] is either in [Formula: see text] or has at least one neighbor in [Formula: see text]. The minimum cardinality of such a set is called the domination number of [Formula: see text], [Formula: see text] [Formula: see text] strongly dominates [Formula: see text] and [Formula: see text] weakly dominates [Formula: see text] if (i) [Formula: see text] and (ii) [Formula: see text] A set [Formula: see text] is a strong-dominating set, shortly sd-set, (weak-dominating set, shortly wd-set) of [Formula: see text] if every vertex in [Formula: see text] is strongly (weakly) dominated by at least one vertex in [Formula: see text]. The strong (weak) domination number [Formula: see text] of [Formula: see text] is the minimum cardinality of an sd-set (wd-set). In this paper, we present weak and strong domination numbers of thorn graphs.

Let [Formula: see text] be a quasi-compact scheme and [Formula: see text]. By [Formula: see text] and [Formula: see text], we denote the set of closed points of [Formula: see text] and the closure of the subset [Formula: see text]. Let [Formula: see text] be a nonempty subset of [Formula: see text]. We define the [Formula: see text]-Zariski topology graph on the scheme [Formula: see text], denoted by [Formula: see text], as an undirected graph whose vertex set is the set [Formula: see text], for two distinct vertices [Formula: see text] and [Formula: see text], there is an arc from [Formula: see text] to [Formula: see text], denoted by [Formula: see text], whenever [Formula: see text]. In this paper, we study the connectivity properties of the graph [Formula: see text], we establish the relationship between the connectivity of the graph [Formula: see text] and the structure of irreducible components of the scheme [Formula: see text]. Also, we characterize when the complement graph of the Zariski topology graph [Formula: see text] is a complete multipartite graph.

An edge irregular total [Formula: see text]-labeling on simple and undirected graph [Formula: see text] is a map [Formula: see text] such that for any different edge [Formula: see text] and [Formula: see text] their weights [Formula: see text] and [Formula: see text] are distinct. The minimum [Formula: see text] for which the graph [Formula: see text] has an edge irregular total [Formula: see text]-labeling is called the total edge irregularity strength of [Formula: see text] and is denoted by tes[Formula: see text]. In this paper, we determine the exact value of the total edge irregularity strength of families of ladder-related graphs, namely triangular ladder graphs, diagonal ladder graphs and circular triangular ladder graphs.

Let [Formula: see text] be a fixed integer, we define the multiplicative function [Formula: see text], where [Formula: see text] is the Piltz divisor function and [Formula: see text] is the unitary analogue function of [Formula: see text]. The main purpose of this paper to use elementary methods to study the mean value of the function [Formula: see text].