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In this paper, we have proved that minimum condition in a pseudo-ordered set (psoset) is equivalent to the descending pseudo-chain condition. Characterization of a pseudo-chain in an acyclic psoset is obtained using graph theoretic approach as transitivity need not hold in psosets. Skala proved that every complete trellis has Fixed Point Property. A counterexample given in this paper shows that trellis having Fixed Point Property need not be complete. The notion of weakly isotone mapping and Strong Fixed Point Property is introduced in a psoset and the characterization of weakly isotone mapping is obtained. It is proved that a connected psoset containing a nontrivial cycle does not have Strong Fixed Point Property.

In this paper, we introduce a new operator in order to derive some new symmetric properties of [Formula: see text]-Fibonacci and [Formula: see text]-Lucas numbers and Fibonacci polynomials. By making use of the new operator defined in this paper, we give some new generating functions for [Formula: see text]-Fibonacci and Pell numbers and Fibonacci polynomials.

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.

Let $a,b,c $ be any positive integers such that $c\mid ab$ and $d_i^\pm$ is a square free positive integer of the form $d_i^\pm=a^{2k} b^{2l}\pm i c^m$ where $k,l \geq m$ and $i=1,2.$ The main focus of this paper to find the fundamental solution of the equation $ x^2-d_i^\pm y^2=1$ with the help of the continued fraction of $\sqrt{d_i^\pm}.$ We also obtain all the positive solutions of the equations $ x^2-d_i^\pm y^2=\pm 1$ and $ x^2-d_i^\pm y^2=\pm 4$ by means of the Fibonacci and Lucas sequences. Furthermore, in this work, we derive some algebraic relations on the Pell form $ F_{d_i^\pm}(x, y) = x^2-d_i^\pm y^2 $ including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation $ F_{\Delta_{d_i^\pm}} (x, y) = 1 $ in terms of $d_i^\pm. We generalized all the results of the papers [2], [9], [26], and [37]., Comment: 16 pages

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*.

In [M. Chandramouleeswaran and P. Muralikrishna, The recipotent abelian group, Int. J. Math. Sci. 5(4) (2010) 191–200], the concept of a recipotent matrix was introduced. A matrix [Formula: see text] of order [Formula: see text] is called recipotent matrix if [Formula: see text] where [Formula: see text] is reversible matrix of [Formula: see text], defined by each nonzero element of [Formula: see text] is replaced by its reciprocal. The aim of this paper is to improve and complete that paper.

In this paper, we study the groups whose subgroups are either almost normal or contranormal. The main theorem of the paper describes locally soluble groups with this property. We also provide a description of locally soluble group, whose subgroups are almost normal or abnormal.

In this paper, we prove the existence of best proximity point and coupled best proximity point on metric spaces with partial order for weak proximal contraction mappings such that these critical points satisfy some constraint inequalities.

In this paper, we classify the gyrotransversals upto isomorphism in a group [Formula: see text] to a fixed subgroup of it. As an application, we have calculated the isomorphism class of gyrotransversals in finite dihedral group to any subgroup of it. As a consequence, we get a lower bound for non-isomorphic right gyrogroups of order [Formula: see text].

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.

Let [Formula: see text] be a group acting on an abelian group [Formula: see text] via a homomorphism [Formula: see text] and let a 2-cocycle [Formula: see text]. By Schreier’s theorem, the pair [Formula: see text] determines a group [Formula: see text] which can arise as a non-split extension of [Formula: see text] by [Formula: see text], denoted by [Formula: see text] and called the perturbed semidirect product of [Formula: see text] by [Formula: see text] under [Formula: see text]. In this paper, we classify the perturbed semidirect products and give some of their properties. Furthermore, we find necessary and sufficient conditions for two perturbed semidirect products to be isomorphic.

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 the [Formula: see text]-dimensional Heisenberg group and let [Formula: see text] be the [Formula: see text]-dimensional torus acting on [Formula: see text] by automorphisms. We consider the semidirect product group [Formula: see text] The cortex, [Formula: see text] of [Formula: see text] is the set of all unitary irreducible representations [Formula: see text] in the unitary dual [Formula: see text] of [Formula: see text] that cannot be Hausdorff separated from the identity representation [Formula: see text] of [Formula: see text] In this paper, we describe explicitly the cortex ([Formula: see text]) of [Formula: see text] using the coadjoint orbits of the group.

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].

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.

In this paper, we investigate the structure of a finite group [Formula: see text] which is totally, mutually or mutually [Formula: see text]-permutable product of proper subgroups [Formula: see text] and [Formula: see text] under theassumption that all maximal subgroups of [Formula: see text] are [Formula: see text]-groups.

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].

A closed knight’s tour of a normal two-dimensional chessboard by using legal moves of the knight has been generalized in several ways. One way is to consider a closed knight’s tour on a ringboard of width [Formula: see text], which is the [Formula: see text] chessboard with the middle part missing and the rim contains [Formula: see text] rows and [Formula: see text] columns. Another way is to stack [Formula: see text] copies of the [Formula: see text] chessboard to construct an [Formula: see text] rectangular chessboard and the closed knight’s tour can be on the surface or within the [Formula: see text] rectangular chessboard. This paper combines these two ideas by stacking [Formula: see text] copies of [Formula: see text] ringboard of width [Formula: see text], which we call the [Formula: see text]-rectangular tube. We explore the existence and the nonexistence of closed knight’s tours for [Formula: see text]-tube and [Formula: see text]-tube.

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 [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]

We consider the skew Laplacian matrix of a digraph [Formula: see text] obtained by giving an arbitrary direction to the edges of a graph [Formula: see text] having [Formula: see text] vertices and [Formula: see text] edges. With [Formula: see text] to be the skew Laplacian eigenvalues of [Formula: see text], the skew Laplacian energy [Formula: see text] of [Formula: see text] is defined as [Formula: see text]. In this paper, we analyze the effect of changing the orientation of an induced subdigraph on the skew Laplacian spectrum. We obtain bounds for the skew Laplacian energy [Formula: see text] in terms of various parameters associated with the digraph [Formula: see text] and the underlying graph [Formula: see text] and we characterize the extremal digraphs attaining these bounds. We also show these bounds improve some known bounds for some families of digraphs. Further, we show the existence of some families of skew Laplacian equienergetic digraphs.

The authors, in 2016, introduced the notion of central autocommutator subgroup of a group [Formula: see text] and proved some new results concerning the central kernel subgroup of [Formula: see text], which was introduced earlier by Haimo in [Formula: see text]. In this paper, we establish some results on central kernel quotient of [Formula: see text]. Finally, we introduce the concept of central autonilpotent (henceforth [Formula: see text]-nilpotent) groups and determine its relationship with the nilpotency of the subgroups of central automorphism of [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.

Let [Formula: see text] be a group and [Formula: see text] be the [Formula: see text]-absolute center of [Formula: see text], that is, the set of all elements of [Formula: see text] fixed by all class preserving automorphisms of [Formula: see text]. In this paper, we classify all finite groups [Formula: see text], whose [Formula: see text]-absolute central factors are isomorphic to the direct product of cyclic groups, [Formula: see text] and [Formula: see text]. Moreover, we consider finite groups which can be written as the union of centralizers of class preserving automorphisms and study the structure of [Formula: see text] for groups, in which the number of distinct centralizers of class preserving automorphisms is equal to 4 or 5.

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.

The Wiener index [Formula: see text] of a connected graph [Formula: see text] is the sum of distances between all pairs of vertices of [Formula: see text]. A connected graph [Formula: see text] is said to be a cactus if each of its blocks is either a cycle or an edge. Let [Formula: see text] be the set of all [Formula: see text]-vertex cacti containing exactly [Formula: see text] cycles. Liu and Lu (2007) determined the unique graph in [Formula: see text] with the minimum Wiener index. Gutman, Li and Wei (2017) determined the unique graph in [Formula: see text] with maximum Wiener index. In this paper, we present the second-minimum Wiener index of graphs in [Formula: see text] and identify the corresponding extremal graphs, which solve partially the problem proposed by Gutman et al. [Cacti with [Formula: see text]-vertices and [Formula: see text] cycles having extremal Wiener index, Discrete Appl. Math. 232 (2017) 189–200] in 2017.

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.

In [Kar, Purkait and Sarkar, Birkhoff center of [Formula: see text]-semiring, Asian-Europ. J. Math. 12(1) (2019) 1950003, 11 pp.], Kar et al. introduced the notion of Birkhoff center of a [Formula: see text]-semiring and obtained some characterizations of this center. In this paper, we study different types of lattice structures of Birkhoff center of a [Formula: see text]-semiring. We also prove that if two [Formula: see text]-semirings are isomorphic, then their Birkhoff centers are isomorphic but the converse is not necessarily true.

In this paper, we introduce and study a new class of rings which we call semireversible rings. A ring [Formula: see text] is called semireversible if for any [Formula: see text] implies there exists a positive integer [Formula: see text] such that [Formula: see text]. We observe that the class of semireversible rings strictly lies between the class of central reversible rings and weakly reversible rings. Some relations are provided between semireversible rings and many other known classes of rings. Some extensions of semireversible rings such as ring of fractions, Dorroh extension, subrings of matrix rings are investigated. Finally, we study semireversible rings via modules over them wherein among other results, we prove that a semireversible left (right) SF-ring is strongly regular.

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 1904, Schur proved his famous result which says that if the central factor group of a given group [Formula: see text] is finite, then so is its derived subgroup. In 1994, Hegarty showed that if the absolute central factor group, [Formula: see text], is finite, then so is its autocommutator subgroup, [Formula: see text]. In this paper, we introduce the concept of left and right [Formula: see text]-commutator, [Formula: see text], and [Formula: see text], where [Formula: see text] is an automorphism of the group [Formula: see text]. Then under some condition, we prove that the finiteness of [Formula: see text] implies that [Formula: see text] is also finite. We also construct an upper bound for the order of [Formula: see text] in terms of the order of [Formula: see text].

The inverse sum indeg index of a graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the degree of the vertex [Formula: see text]. In a recent paper, Pattabiraman [Inverse sum indeg index of graphs, AKCE Int. J. Graphs Combinat. 15(2) (2018) 155–167] gave some lower and upper bounds on [Formula: see text] index of all connected graphs in terms of Harmonic index, second Zagreb index and hyper Zagreb index. But some results were erroneous. In this note, we have corrected these results.

In this paper, first Zagreb index, second Zagreb index, first multiplicative Zagreb index, second multiplicative Zagreb index, first Zagreb coindices index, second Zagreb coindices index, first multiplicative Zagreb coindices index, second multiplicative Zagreb coindices index of [Formula: see text] have been established, where [Formula: see text] and [Formula: see text] are prime.

The concept of zero-divisor graph of a commutative ring was introduced by Beck [Coloring of commutating ring, J. Algebra 116 (1988) 208–226]. In this paper, we present some properties of zero divisor graphs obtained from ring [Formula: see text], where [Formula: see text] and [Formula: see text] are primes. Also, we give some degree-based topological indices of this special graph.

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.

The focus of this paper is to study the two-variable Kauffman polynomials [Formula: see text] and [Formula: see text], and the one-variable BLM/Ho polynomial [Formula: see text] of [Formula: see text]-torus link as the Fibonacci-type polynomials and to express the Kauffman polynomials in terms of the BLM/Ho polynomial. For this purpose, we prove that each of the examined polynomials of [Formula: see text]-torus link can be determined by a third-order recurrence relation and give the recursive properties of them. We correlate these polynomials with the Fibonacci-type polynomials. By using the relations between the BLM/Ho polynomials and Fibonacci-type polynomials, we express the Kauffman polynomials in terms of the BLM/Ho polynomials.

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.