Let G be a graph. The inherent Inj-equitable graph of a graph G (IIE(G)) is the graph with the sa... more Let G be a graph. The inherent Inj-equitable graph of a graph G (IIE(G)) is the graph with the same vertices as G and any two vertices u and v are adjacent in IIE(G) if they are adjacent in G and |deg in (u) − deg in (v)| 1, where for any vertex w ∈ V (G), deg in (w) = {w ′ ∈ V : N (w ′) ∩ N (w) ̸ = ϕ} [2]. In this paper, inherent Inj-equitable graph of some graphs are obtained, some properties and results are established. We define iterated Inj-equitable graph of a graph, complete Inj-equitable graph and we define the Inj-equitable graph.
International Journal of Analysis and Applications, 2022
One of the tools, to research and investigation the structural dependence of various properties a... more One of the tools, to research and investigation the structural dependence of various properties and some activities of chemical structures and networks is the topological indices of graphs. In this research work, we introduce novel indices of graphs which they based on the uphill degree of the vertices termed as uphill Zagreb topological indices. Exact formulae of these new indices for some important and famous families of graphs are established.
Let m and n be fixed integers with m ≥ 1, n ≥ 1 and let R be a (m + n + 2)!− torsion free semipri... more Let m and n be fixed integers with m ≥ 1, n ≥ 1 and let R be a (m + n + 2)!− torsion free semiprime ring with the identity element suppose there exists an additive mapping T : R → R such that T (x m+n+1) = θ(x m)T (x)θ(x n) holds for all x ∈ R where θ is an automorphisms of R. In this case T is a centralizer.
The original version of this article unfortunately missed to include another Acknowledgments belo... more The original version of this article unfortunately missed to include another Acknowledgments below. "The authors acknowledge the financial support provided by the NSTIP strategic technologies program in the Kingdom of Saudi Arabia-Project No (12-MAT3055-03), and extend the thanks to the Science and Technology Unit, King Abdulaziz University for their technical support." The original paper was updated.
International Journal of Analysis and Applications, 2020
For a graph G = (V, E), a vertex coloring (or, simply, a coloring) of G is a function C : V (G) →... more For a graph G = (V, E), a vertex coloring (or, simply, a coloring) of G is a function C : V (G) → {1, 2, ..., k} (using the non-negative integers {1, 2, ..., k} as colors). In this research work, we introduce a new type of graph coloring called monotone coloring, along with this new coloring, we define the monotone chromatic number of a graph and establish some related new graphs. Basic properties and exact values of the monotone chromatic number of some graph families, like standard graphs, Kragujevac trees and firefly graph are obtained. Also, we get a characterization for bipartite graphs by defining the monotone bipartite graph. Exact values of the monotone chromatic number for some special case of Cartesian product of graphs are found. Finally, upper and lower bounds for monotone chromatic number of the Cartesian product for non trivial connected graphs are presented.
Let [Formula: see text] be a finite undirected graph without loops or multiple edges. A non-empty... more Let [Formula: see text] be a finite undirected graph without loops or multiple edges. A non-empty set of vertices [Formula: see text] is called defensive alliance if every vertex in [Formula: see text] have at least one more neighbors inside of [Formula: see text] than it has outside of [Formula: see text]. A non-empty set of vertices [Formula: see text] is called a dominating set if every vertex not in [Formula: see text] is adjacent to at least one member of [Formula: see text]. A defensive alliance dominating set is called global. The global defensive alliance number [Formula: see text] is defined as the minimum cardinality among all global defensive alliances. In this paper, we initiate the study of the global defensive alliance number of zero-divisor graphs [Formula: see text] with [Formula: see text] is a finite commutative ring. Hence, we calculate [Formula: see text] for some usual kind of rings. We finish by a complete characterization of rings with [Formula: see text].
Abstract Our purpose in this paper is to show that certain subsets, defined by commutativity cond... more Abstract Our purpose in this paper is to show that certain subsets, defined by commutativity conditions involving derivations or endomorphisms, coincide with the center in prime rings. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.
This short exposition is about some commutativity conditions on a semiprime right Goldie C k-ring... more This short exposition is about some commutativity conditions on a semiprime right Goldie C k-ring. In particular, it is observed here that a semiprime right Goldie C k-ring with symmetric quotient is commutative. The statement holds if the symmetric ring is replaced by reduced, 2-primal, left duo, right duo, abelian, NI, NCI, IFP, or Armendariz ring.
Let R be a ring with center Z(R). A mapping H from R into itself is called a centrally-extended h... more Let R be a ring with center Z(R). A mapping H from R into itself is called a centrally-extended homoderivation on R if for each x, y ∈ R, H(x + y) − H(x) − H(y) ∈ Z(R) and H(xy) − H(x)H(y) − H(x)y − xH(y) ∈ Z(R). We present examples of mappings that are centrally-extended homoderivations but not homoderivations. We also study the effect of such mappings on Z(R) and prove some commutativity results.
The main objective of the present paper is to prove the following result: let and be positive int... more The main objective of the present paper is to prove the following result: let and be positive integers with and let be an torsion free semiprime ring with identity element. Let be an automorphism of R. Suppose there exists an additive mapping such that for all , then is a derivation on .
Let G be a graph. The inherent Inj-equitable graph of a graph G (IIE(G)) is the graph with the sa... more Let G be a graph. The inherent Inj-equitable graph of a graph G (IIE(G)) is the graph with the same vertices as G and any two vertices u and v are adjacent in IIE(G) if they are adjacent in G and jdegin(u)
Let R be a ring with center Z and α, β and d mappings of R. A mapping F of R is called a centrall... more Let R be a ring with center Z and α, β and d mappings of R. A mapping F of R is called a centrally-extended multiplicative (generalized)-(α, β)-derivation associated with d if F (xy) − F (x)α(y) − β(x)d(y) ∈ Z for all x, y ∈ R. The objective of the present paper is to study the following conditions: (i) F (xy) ± β(x)G(y) ∈ Z, (ii) F (xy) ± g(x)α(y) ∈ Z and (iii) F (xy) ± g(y)α(x) ∈ Z for all x, y in some appropriate subsets of R, where G is a multiplicative (generalized)-(α, β)-derivation of R associated with the map g on R.
We analyse the normal quotient structure of several infinite families of finite connected edge-tr... more We analyse the normal quotient structure of several infinite families of finite connected edge-transitive, four-valent oriented graphs. These families were singled out by Marušič and others to illustrate various different internal structures for these graphs in terms of their alternating cycles (cycles in which consecutive edges have opposite orientations). Studying the normal quotients gives fresh insights into these oriented graphs: in particular we discovered some unexpected 'cross-overs' between these graph families when we formed normal quotients. We determine which of these oriented graphs are 'basic', in the sense that their only proper normal quotients are degenerate. Moreover, we show that the three types of edge-orientations studied are the only orientations, of the underlying undirected graphs in these families, which are invariant under a group action which is both vertex-transitive and edge-transitive.
Our purpose in this paper is to investigate some particular classes of generalized derivations an... more Our purpose in this paper is to investigate some particular classes of generalized derivations and their relationship with commutativity of prime rings with involution. Some well-known results characterizing commutativity of prime rings have been generalized. Furthermore, we provide examples to show that the assumed restrictions cannot be relaxed.
Let R be a prime ring with center Z(R). A map G : R →R is called a multiplicative (generalized) (... more Let R be a prime ring with center Z(R). A map G : R →R is called a multiplicative (generalized) (α, β)-derivation if G(xy)= G(x)α(y)+β(x)g(y) is fulfilled for all x; y ∈ R, where g : R → R is any map (not necessarily derivation) and α; β : R → R are automorphisms. Suppose that G and H are two multiplicative (generalized) (α, β)-derivations associated with the mappings g and h, respectively, on R and α, β are automorphisms of R. The main objective of the present paper is to investigate the following algebraic identities: (i) G(xy) + α(xy) = 0, (ii) G(xy) + α(yx) = 0, (iii) G(xy) + G(x)G(y) = 0, (iv) G(xy) = α(y) ○ H(x) and (v) G(xy) = [α(y), H(x)] for all x, y in an appropriate subset of R.
Sum of two nilpotent elements in a ring may not be nilpotent in general, but for commutative ring... more Sum of two nilpotent elements in a ring may not be nilpotent in general, but for commutative rings this sum is nilpotent. In between commutative and non-commutative rings there are several types of rings in which this property holds. For instance, reduced, NI, AI (or IFP), 2-primal, reversible and symmetric, etc. We may term these types of rings as nearby commutative rings (in short NC-rings). In this work we have studied properties and various characterizations of such rings as well as rngs. As applications, we have investigated some commutativity conditions by involving semi-projective-Morita-contexts and right C k-Goldie rings.
We develop a new framework for analysing finite connected, oriented graphs of valency 4, which ad... more We develop a new framework for analysing finite connected, oriented graphs of valency 4, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a subfamily of 'basic' graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective further analysis, and we demonstrate that this is so for the quasiprimitive basic graphs. Here we obtain strong restirictions on the group involved, and construct several infinite families of such graphs which, to our knowledge, are different from any recorded in the literature so far. Several open problems are posed in the paper.
Let G be a graph. The inherent Inj-equitable graph of a graph G (IIE(G)) is the graph with the sa... more Let G be a graph. The inherent Inj-equitable graph of a graph G (IIE(G)) is the graph with the same vertices as G and any two vertices u and v are adjacent in IIE(G) if they are adjacent in G and |deg in (u) − deg in (v)| 1, where for any vertex w ∈ V (G), deg in (w) = {w ′ ∈ V : N (w ′) ∩ N (w) ̸ = ϕ} [2]. In this paper, inherent Inj-equitable graph of some graphs are obtained, some properties and results are established. We define iterated Inj-equitable graph of a graph, complete Inj-equitable graph and we define the Inj-equitable graph.
International Journal of Analysis and Applications, 2022
One of the tools, to research and investigation the structural dependence of various properties a... more One of the tools, to research and investigation the structural dependence of various properties and some activities of chemical structures and networks is the topological indices of graphs. In this research work, we introduce novel indices of graphs which they based on the uphill degree of the vertices termed as uphill Zagreb topological indices. Exact formulae of these new indices for some important and famous families of graphs are established.
Let m and n be fixed integers with m ≥ 1, n ≥ 1 and let R be a (m + n + 2)!− torsion free semipri... more Let m and n be fixed integers with m ≥ 1, n ≥ 1 and let R be a (m + n + 2)!− torsion free semiprime ring with the identity element suppose there exists an additive mapping T : R → R such that T (x m+n+1) = θ(x m)T (x)θ(x n) holds for all x ∈ R where θ is an automorphisms of R. In this case T is a centralizer.
The original version of this article unfortunately missed to include another Acknowledgments belo... more The original version of this article unfortunately missed to include another Acknowledgments below. "The authors acknowledge the financial support provided by the NSTIP strategic technologies program in the Kingdom of Saudi Arabia-Project No (12-MAT3055-03), and extend the thanks to the Science and Technology Unit, King Abdulaziz University for their technical support." The original paper was updated.
International Journal of Analysis and Applications, 2020
For a graph G = (V, E), a vertex coloring (or, simply, a coloring) of G is a function C : V (G) →... more For a graph G = (V, E), a vertex coloring (or, simply, a coloring) of G is a function C : V (G) → {1, 2, ..., k} (using the non-negative integers {1, 2, ..., k} as colors). In this research work, we introduce a new type of graph coloring called monotone coloring, along with this new coloring, we define the monotone chromatic number of a graph and establish some related new graphs. Basic properties and exact values of the monotone chromatic number of some graph families, like standard graphs, Kragujevac trees and firefly graph are obtained. Also, we get a characterization for bipartite graphs by defining the monotone bipartite graph. Exact values of the monotone chromatic number for some special case of Cartesian product of graphs are found. Finally, upper and lower bounds for monotone chromatic number of the Cartesian product for non trivial connected graphs are presented.
Let [Formula: see text] be a finite undirected graph without loops or multiple edges. A non-empty... more Let [Formula: see text] be a finite undirected graph without loops or multiple edges. A non-empty set of vertices [Formula: see text] is called defensive alliance if every vertex in [Formula: see text] have at least one more neighbors inside of [Formula: see text] than it has outside of [Formula: see text]. A non-empty set of vertices [Formula: see text] is called a dominating set if every vertex not in [Formula: see text] is adjacent to at least one member of [Formula: see text]. A defensive alliance dominating set is called global. The global defensive alliance number [Formula: see text] is defined as the minimum cardinality among all global defensive alliances. In this paper, we initiate the study of the global defensive alliance number of zero-divisor graphs [Formula: see text] with [Formula: see text] is a finite commutative ring. Hence, we calculate [Formula: see text] for some usual kind of rings. We finish by a complete characterization of rings with [Formula: see text].
Abstract Our purpose in this paper is to show that certain subsets, defined by commutativity cond... more Abstract Our purpose in this paper is to show that certain subsets, defined by commutativity conditions involving derivations or endomorphisms, coincide with the center in prime rings. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.
This short exposition is about some commutativity conditions on a semiprime right Goldie C k-ring... more This short exposition is about some commutativity conditions on a semiprime right Goldie C k-ring. In particular, it is observed here that a semiprime right Goldie C k-ring with symmetric quotient is commutative. The statement holds if the symmetric ring is replaced by reduced, 2-primal, left duo, right duo, abelian, NI, NCI, IFP, or Armendariz ring.
Let R be a ring with center Z(R). A mapping H from R into itself is called a centrally-extended h... more Let R be a ring with center Z(R). A mapping H from R into itself is called a centrally-extended homoderivation on R if for each x, y ∈ R, H(x + y) − H(x) − H(y) ∈ Z(R) and H(xy) − H(x)H(y) − H(x)y − xH(y) ∈ Z(R). We present examples of mappings that are centrally-extended homoderivations but not homoderivations. We also study the effect of such mappings on Z(R) and prove some commutativity results.
The main objective of the present paper is to prove the following result: let and be positive int... more The main objective of the present paper is to prove the following result: let and be positive integers with and let be an torsion free semiprime ring with identity element. Let be an automorphism of R. Suppose there exists an additive mapping such that for all , then is a derivation on .
Let G be a graph. The inherent Inj-equitable graph of a graph G (IIE(G)) is the graph with the sa... more Let G be a graph. The inherent Inj-equitable graph of a graph G (IIE(G)) is the graph with the same vertices as G and any two vertices u and v are adjacent in IIE(G) if they are adjacent in G and jdegin(u)
Let R be a ring with center Z and α, β and d mappings of R. A mapping F of R is called a centrall... more Let R be a ring with center Z and α, β and d mappings of R. A mapping F of R is called a centrally-extended multiplicative (generalized)-(α, β)-derivation associated with d if F (xy) − F (x)α(y) − β(x)d(y) ∈ Z for all x, y ∈ R. The objective of the present paper is to study the following conditions: (i) F (xy) ± β(x)G(y) ∈ Z, (ii) F (xy) ± g(x)α(y) ∈ Z and (iii) F (xy) ± g(y)α(x) ∈ Z for all x, y in some appropriate subsets of R, where G is a multiplicative (generalized)-(α, β)-derivation of R associated with the map g on R.
We analyse the normal quotient structure of several infinite families of finite connected edge-tr... more We analyse the normal quotient structure of several infinite families of finite connected edge-transitive, four-valent oriented graphs. These families were singled out by Marušič and others to illustrate various different internal structures for these graphs in terms of their alternating cycles (cycles in which consecutive edges have opposite orientations). Studying the normal quotients gives fresh insights into these oriented graphs: in particular we discovered some unexpected 'cross-overs' between these graph families when we formed normal quotients. We determine which of these oriented graphs are 'basic', in the sense that their only proper normal quotients are degenerate. Moreover, we show that the three types of edge-orientations studied are the only orientations, of the underlying undirected graphs in these families, which are invariant under a group action which is both vertex-transitive and edge-transitive.
Our purpose in this paper is to investigate some particular classes of generalized derivations an... more Our purpose in this paper is to investigate some particular classes of generalized derivations and their relationship with commutativity of prime rings with involution. Some well-known results characterizing commutativity of prime rings have been generalized. Furthermore, we provide examples to show that the assumed restrictions cannot be relaxed.
Let R be a prime ring with center Z(R). A map G : R →R is called a multiplicative (generalized) (... more Let R be a prime ring with center Z(R). A map G : R →R is called a multiplicative (generalized) (α, β)-derivation if G(xy)= G(x)α(y)+β(x)g(y) is fulfilled for all x; y ∈ R, where g : R → R is any map (not necessarily derivation) and α; β : R → R are automorphisms. Suppose that G and H are two multiplicative (generalized) (α, β)-derivations associated with the mappings g and h, respectively, on R and α, β are automorphisms of R. The main objective of the present paper is to investigate the following algebraic identities: (i) G(xy) + α(xy) = 0, (ii) G(xy) + α(yx) = 0, (iii) G(xy) + G(x)G(y) = 0, (iv) G(xy) = α(y) ○ H(x) and (v) G(xy) = [α(y), H(x)] for all x, y in an appropriate subset of R.
Sum of two nilpotent elements in a ring may not be nilpotent in general, but for commutative ring... more Sum of two nilpotent elements in a ring may not be nilpotent in general, but for commutative rings this sum is nilpotent. In between commutative and non-commutative rings there are several types of rings in which this property holds. For instance, reduced, NI, AI (or IFP), 2-primal, reversible and symmetric, etc. We may term these types of rings as nearby commutative rings (in short NC-rings). In this work we have studied properties and various characterizations of such rings as well as rngs. As applications, we have investigated some commutativity conditions by involving semi-projective-Morita-contexts and right C k-Goldie rings.
We develop a new framework for analysing finite connected, oriented graphs of valency 4, which ad... more We develop a new framework for analysing finite connected, oriented graphs of valency 4, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a subfamily of 'basic' graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective further analysis, and we demonstrate that this is so for the quasiprimitive basic graphs. Here we obtain strong restirictions on the group involved, and construct several infinite families of such graphs which, to our knowledge, are different from any recorded in the literature so far. Several open problems are posed in the paper.
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