Let $G=(V,E)$ be a graph. The injective neighborhood of a vertex $u\in V(G)$ denoted by $N_{in}(u... more Let $G=(V,E)$ be a graph. The injective neighborhood of a vertex $u\in V(G)$ denoted by $N_{in}(u)$ is defined as $N_{in}(u)=\{v\in V(G):|\Gamma(u,v)|\geq 1\}$, where $|\Gamma(u,v)|$ is the number of common neighborhoods between the vertices $u$ and $v$ in $G$. The cardinality of $N_{in}(u)$ is called the injective degree of the vertex $u$ in $G$ and denoted by $deg_{in}(u)$, \cite{20}. In this paper, we introduce the injective Zagreb indices of a graph $G$ as $M_1^{inj}(G)=\sum_{u\in V(G)}\big[deg_{in}(u)\big]^2$, $M_2^{inj}(G)=\sum_{uv\in E(G)}deg_{in}(u)deg_{in}(v)$, respectively, and the relative injective Zagreb indices as $RM_1^{inj}(G)=\sum_{u\in V(G)}deg_{in}(u)deg(u)$, $RM_2^{inj}(G)=\sum_{uv\in E(G)}\big[deg_{in}(u)deg(v)+deg(u)deg_{in}(v)\big]$, respectively. Some properties of these topological indices are obtained. Exact values for some families of graphs and some graph operations are computed.
Given a graph $G = (V,E)$, a dominating set $D subseteq� V$ is called a semi-strong split dominat... more Given a graph $G = (V,E)$, a dominating set $D subseteq� V$ is called a semi-strong split dominating set of $G$ if $|V setminus D| geq� 1$ and the maximum degree of the subgraph induced by $V setminus D$ is 1. The minimum cardinality of a semi-strong split dominating set (SSSDS) of G is the semi-strong split domination number of G, denoted $gamma_{sss}(G)$. In this work, we introduce the concept and prove several results regarding it.
In this paper, we introduce the concept of co-common neighborhood domination number (CCN-dominati... more In this paper, we introduce the concept of co-common neighborhood domination number (CCN-domination number) γccn(G) of a graph G and we study its relation with the standard domination number γ(G). We also define CCN-independence number βccn(G), total CCNdomination number γtccn(G), CCN-covering number αccn(G) and CCN-domatic number.
In this paper, we introduce a new type of graph energy called the non-common-neighborhood energy ... more In this paper, we introduce a new type of graph energy called the non-common-neighborhood energy () E G NCN , NCN-energy for some standard graphs is obtained and an upper bound for () E G NCN is found when G is a strongly regular graph. Also the relation between common neighbourhood energy and non-common neighbourhood energy of a graph is established.
By the degree, equitable adjacency between vertices can be redefine almost all of the variants of... more By the degree, equitable adjacency between vertices can be redefine almost all of the variants of the graphs. In this paper we study the degree equitability of the graph by defining equitable connectivity, equitable regularity, equitable connected graph and equitable complete graph. Some new families of graphs and some interesting results are obtained.
International journal of computer applications, Sep 15, 2016
The near common-neighborhood graph of a graph G, denoted by ncn(G), is the graph on the same vert... more The near common-neighborhood graph of a graph G, denoted by ncn(G), is the graph on the same vertices of G, two vertices being adjacent in ncn(G) if there is at least one vertex in G not adjacent to both of them. A graph is called near-common neighborhood graph if it is the near-common neighborhood of some graph. In this paper we introduce the near-common neighborhood of a graph, the near common neighborhood graph, near-completeness number of a graph, basic properties of these new graphs are obtained and interesting results are established.
International Journal of Analysis and Applications, 2020
Topological indices and domination in graphs are the essential topics in the theory of graphs. A ... more Topological indices and domination in graphs are the essential topics in the theory of graphs. A set of vertices D ⊆ V (G) is said to be a dominating set for G if any vertex v ∈ V − D is adjacent to some vertex u ∈ D. In this research work, we define a new degree of each vertex v ∈ V (G), called the domination degree of v and denoted by dd(v), along with this new degree some domination indices based on domination degree are introduced. We study some basic properties of the domination degree function. Exact values and bounds for domination Zagreb indices of some families of graphs including the join and corona product are obtained. Finally, we generalize the domination degree of the vertex and new general indices are defined.
In this paper, for every k-degree distance we define ψk-Polynomial of a connected graph G=(V,E) a... more In this paper, for every k-degree distance we define ψk-Polynomial of a connected graph G=(V,E) as where ψk(i,j) is the number of edges uv in G such that {dk(v), dk(u)}={i,j} and dk(v), dk(u) are the kth degree distance of v and u, and δk, ∆k are the minimum and maximum kth distance degrees respectively. We compute the ψ2-Polynomial of some standard graphs and some graph operations. Also ψ2-Polynomial for honeycomb network and Graphene are obtained with their plotting in 3D.
Let G=(V,E)be a graph. A subset D of V is called common neighbourhood dominating set (CN-dominati... more Let G=(V,E)be a graph. A subset D of V is called common neighbourhood dominating set (CN-dominating set) if for every v∈V−D there exists a vertex u∈D such that uv∈E(G) and |Γ(u,v)|≥1, where |Γ(u,v)| is the number of common neighbourhood between the verticesu and v. The minimum cardinality of such CN-dominating set denoted by γcn(G) and is called common neighbourhood domination number (CN-edge domination).
Discrete Mathematics, Algorithms and Applications, Jun 1, 2018
The Zagreb indices have been introduced in 1972 to explain some properties of chemical compounds ... more The Zagreb indices have been introduced in 1972 to explain some properties of chemical compounds at molecular level mathematically. Since then, the Zagreb indices have been studied extensively due to their ease of calculation and their numerous applications in place of the existing chemical methods which needed more time and increased the costs. Many new kinds of Zagreb indices are recently introduced for several similar reasons. In this paper, we introduce the entire Zagreb indices by adding incidency of edges and vertices to the adjacency of the vertices. Our motivation in doing so was the following fact about molecular graphs: The intermolecular forces do not only exist between the atoms, but also between the atoms and bonds, so one should also take into account the relations (forces) between edges and vertices in addition to the relations between vertices to obtain better approximations to intermolecular forces. Exact values of these indices for some families of graphs are obtained and some important properties of the entire Zagreb indices are established.
Bulletin Classe des sciences mathematiques et natturalles
We introduce the concept of common-neighborhood energy E CN of a graph G and obtain an upper boun... more We introduce the concept of common-neighborhood energy E CN of a graph G and obtain an upper bound for E CN when G is strongly regular. We also show that E CN of several classes of graphs is less than the common-neighborhood energy of the complete graph K n .
The Zagreb indices have been introduced more than forty four years ago by Gutman and Trinajestic ... more The Zagreb indices have been introduced more than forty four years ago by Gutman and Trinajestic as the sum of the squares of the degrees of the vertices, and the sum of the products of the degrees of pairs of adjacent ver-tices, respectively, [7]. In this paper, we introduce the first and second equitable and non-equitable Zagreb indices as M e 1 (G) = u∈V (G)
Let G = (V, E) be a connected graph. A subset S of V is called injective dominating set (Inj-domi... more Let G = (V, E) be a connected graph. A subset S of V is called injective dominating set (Inj-dominating set) if for every vertex v ∈ V −S there exists a vertex u ∈ S such that |Γ(u, v)| 1, where |Γ(u, v)| is the number of common neighbors between the vertices u and v. In this research work, we introduce the connected injective domination of graphs. Exact values for some families of graphs, relations with the other domination parameters are obtained. Bounds and some interesting results are established.
In this paper, we introduce the concept of co-common neighborhood domination number (CCN-dominati... more In this paper, we introduce the concept of co-common neighborhood domination number (CCN-domination number) γccn(G) of a graph G and we study its relation with the standard domination number γ(G). We also define CCN-independence number βccn(G), total CCN-domination number γ tccn (G), CCN-covering number αccn(G) and CCN-domatic number.
The independent domination polynomials of some standard graphs are obtained and some properties o... more The independent domination polynomials of some standard graphs are obtained and some properties of the independent domination polynomial of graphs are established.
Graph polynomials are polynomials associated to graphs that encode the number of subgraphs with g... more Graph polynomials are polynomials associated to graphs that encode the number of subgraphs with given properties. In this paper we introduce a new type of graph polynomial called neighbourhood polynomial. We obtained the neighbourhood polynomial of some interested standard graphs like complete graph, complete bipartite graph, bi-star graph, spider, wounded spider graph and for some corona product graphs.
An injective dominating set S in a graph G is called independent injective dominating set (IInj-d... more An injective dominating set S in a graph G is called independent injective dominating set (IInj-dominating set) if for every v ∈ S, N (v) ∩ S = φ. The minimum cardinality of such injective dominating set is called independent injective domination number of G and denoted by γ i i n (G). In this paper, we introduce the independent injective domination of graphs and we define the IID-graph. Exact values for some families of graphs, relations with the other domination parameters are obtained. Also, we introduce the independent injective frustration number of graphs. Bounds and some interesting results are established.
Vn-Arithmetic graph has been introduced by Vasumathi and
Vangipuram [9]. In this paper some prope... more Vn-Arithmetic graph has been introduced by Vasumathi and Vangipuram [9]. In this paper some properties of Vn-rithmetic graph, maximum degree, minimum degree, number of edges, diameter,radius, Hamiltonian and Eulerian are studied. Also, we introduce m-Arithmetical graphs. Some properties and interesting results for m-Arithmetical graphs are established.
In this paper, we introduce a new type of graph energy called the non-common-neighborhood energy ... more In this paper, we introduce a new type of graph energy called the non-common-neighborhood energy E_{NCN}(G). NCN-energy for some standard graphs is obtained and an upper bound for E_{NCN}(G) is found when G is a strongly regular graph. Also the relation between common neighbourhood energy and non-common neighbourhood energy of a graph is established.
Let $G=(V,E)$ be a graph. The injective neighborhood of a vertex $u\in V(G)$ denoted by $N_{in}(u... more Let $G=(V,E)$ be a graph. The injective neighborhood of a vertex $u\in V(G)$ denoted by $N_{in}(u)$ is defined as $N_{in}(u)=\{v\in V(G):|\Gamma(u,v)|\geq 1\}$, where $|\Gamma(u,v)|$ is the number of common neighborhoods between the vertices $u$ and $v$ in $G$. The cardinality of $N_{in}(u)$ is called the injective degree of the vertex $u$ in $G$ and denoted by $deg_{in}(u)$, \cite{20}. In this paper, we introduce the injective Zagreb indices of a graph $G$ as $M_1^{inj}(G)=\sum_{u\in V(G)}\big[deg_{in}(u)\big]^2$, $M_2^{inj}(G)=\sum_{uv\in E(G)}deg_{in}(u)deg_{in}(v)$, respectively, and the relative injective Zagreb indices as $RM_1^{inj}(G)=\sum_{u\in V(G)}deg_{in}(u)deg(u)$, $RM_2^{inj}(G)=\sum_{uv\in E(G)}\big[deg_{in}(u)deg(v)+deg(u)deg_{in}(v)\big]$, respectively. Some properties of these topological indices are obtained. Exact values for some families of graphs and some graph operations are computed.
Given a graph $G = (V,E)$, a dominating set $D subseteq� V$ is called a semi-strong split dominat... more Given a graph $G = (V,E)$, a dominating set $D subseteq� V$ is called a semi-strong split dominating set of $G$ if $|V setminus D| geq� 1$ and the maximum degree of the subgraph induced by $V setminus D$ is 1. The minimum cardinality of a semi-strong split dominating set (SSSDS) of G is the semi-strong split domination number of G, denoted $gamma_{sss}(G)$. In this work, we introduce the concept and prove several results regarding it.
In this paper, we introduce the concept of co-common neighborhood domination number (CCN-dominati... more In this paper, we introduce the concept of co-common neighborhood domination number (CCN-domination number) γccn(G) of a graph G and we study its relation with the standard domination number γ(G). We also define CCN-independence number βccn(G), total CCNdomination number γtccn(G), CCN-covering number αccn(G) and CCN-domatic number.
In this paper, we introduce a new type of graph energy called the non-common-neighborhood energy ... more In this paper, we introduce a new type of graph energy called the non-common-neighborhood energy () E G NCN , NCN-energy for some standard graphs is obtained and an upper bound for () E G NCN is found when G is a strongly regular graph. Also the relation between common neighbourhood energy and non-common neighbourhood energy of a graph is established.
By the degree, equitable adjacency between vertices can be redefine almost all of the variants of... more By the degree, equitable adjacency between vertices can be redefine almost all of the variants of the graphs. In this paper we study the degree equitability of the graph by defining equitable connectivity, equitable regularity, equitable connected graph and equitable complete graph. Some new families of graphs and some interesting results are obtained.
International journal of computer applications, Sep 15, 2016
The near common-neighborhood graph of a graph G, denoted by ncn(G), is the graph on the same vert... more The near common-neighborhood graph of a graph G, denoted by ncn(G), is the graph on the same vertices of G, two vertices being adjacent in ncn(G) if there is at least one vertex in G not adjacent to both of them. A graph is called near-common neighborhood graph if it is the near-common neighborhood of some graph. In this paper we introduce the near-common neighborhood of a graph, the near common neighborhood graph, near-completeness number of a graph, basic properties of these new graphs are obtained and interesting results are established.
International Journal of Analysis and Applications, 2020
Topological indices and domination in graphs are the essential topics in the theory of graphs. A ... more Topological indices and domination in graphs are the essential topics in the theory of graphs. A set of vertices D ⊆ V (G) is said to be a dominating set for G if any vertex v ∈ V − D is adjacent to some vertex u ∈ D. In this research work, we define a new degree of each vertex v ∈ V (G), called the domination degree of v and denoted by dd(v), along with this new degree some domination indices based on domination degree are introduced. We study some basic properties of the domination degree function. Exact values and bounds for domination Zagreb indices of some families of graphs including the join and corona product are obtained. Finally, we generalize the domination degree of the vertex and new general indices are defined.
In this paper, for every k-degree distance we define ψk-Polynomial of a connected graph G=(V,E) a... more In this paper, for every k-degree distance we define ψk-Polynomial of a connected graph G=(V,E) as where ψk(i,j) is the number of edges uv in G such that {dk(v), dk(u)}={i,j} and dk(v), dk(u) are the kth degree distance of v and u, and δk, ∆k are the minimum and maximum kth distance degrees respectively. We compute the ψ2-Polynomial of some standard graphs and some graph operations. Also ψ2-Polynomial for honeycomb network and Graphene are obtained with their plotting in 3D.
Let G=(V,E)be a graph. A subset D of V is called common neighbourhood dominating set (CN-dominati... more Let G=(V,E)be a graph. A subset D of V is called common neighbourhood dominating set (CN-dominating set) if for every v∈V−D there exists a vertex u∈D such that uv∈E(G) and |Γ(u,v)|≥1, where |Γ(u,v)| is the number of common neighbourhood between the verticesu and v. The minimum cardinality of such CN-dominating set denoted by γcn(G) and is called common neighbourhood domination number (CN-edge domination).
Discrete Mathematics, Algorithms and Applications, Jun 1, 2018
The Zagreb indices have been introduced in 1972 to explain some properties of chemical compounds ... more The Zagreb indices have been introduced in 1972 to explain some properties of chemical compounds at molecular level mathematically. Since then, the Zagreb indices have been studied extensively due to their ease of calculation and their numerous applications in place of the existing chemical methods which needed more time and increased the costs. Many new kinds of Zagreb indices are recently introduced for several similar reasons. In this paper, we introduce the entire Zagreb indices by adding incidency of edges and vertices to the adjacency of the vertices. Our motivation in doing so was the following fact about molecular graphs: The intermolecular forces do not only exist between the atoms, but also between the atoms and bonds, so one should also take into account the relations (forces) between edges and vertices in addition to the relations between vertices to obtain better approximations to intermolecular forces. Exact values of these indices for some families of graphs are obtained and some important properties of the entire Zagreb indices are established.
Bulletin Classe des sciences mathematiques et natturalles
We introduce the concept of common-neighborhood energy E CN of a graph G and obtain an upper boun... more We introduce the concept of common-neighborhood energy E CN of a graph G and obtain an upper bound for E CN when G is strongly regular. We also show that E CN of several classes of graphs is less than the common-neighborhood energy of the complete graph K n .
The Zagreb indices have been introduced more than forty four years ago by Gutman and Trinajestic ... more The Zagreb indices have been introduced more than forty four years ago by Gutman and Trinajestic as the sum of the squares of the degrees of the vertices, and the sum of the products of the degrees of pairs of adjacent ver-tices, respectively, [7]. In this paper, we introduce the first and second equitable and non-equitable Zagreb indices as M e 1 (G) = u∈V (G)
Let G = (V, E) be a connected graph. A subset S of V is called injective dominating set (Inj-domi... more Let G = (V, E) be a connected graph. A subset S of V is called injective dominating set (Inj-dominating set) if for every vertex v ∈ V −S there exists a vertex u ∈ S such that |Γ(u, v)| 1, where |Γ(u, v)| is the number of common neighbors between the vertices u and v. In this research work, we introduce the connected injective domination of graphs. Exact values for some families of graphs, relations with the other domination parameters are obtained. Bounds and some interesting results are established.
In this paper, we introduce the concept of co-common neighborhood domination number (CCN-dominati... more In this paper, we introduce the concept of co-common neighborhood domination number (CCN-domination number) γccn(G) of a graph G and we study its relation with the standard domination number γ(G). We also define CCN-independence number βccn(G), total CCN-domination number γ tccn (G), CCN-covering number αccn(G) and CCN-domatic number.
The independent domination polynomials of some standard graphs are obtained and some properties o... more The independent domination polynomials of some standard graphs are obtained and some properties of the independent domination polynomial of graphs are established.
Graph polynomials are polynomials associated to graphs that encode the number of subgraphs with g... more Graph polynomials are polynomials associated to graphs that encode the number of subgraphs with given properties. In this paper we introduce a new type of graph polynomial called neighbourhood polynomial. We obtained the neighbourhood polynomial of some interested standard graphs like complete graph, complete bipartite graph, bi-star graph, spider, wounded spider graph and for some corona product graphs.
An injective dominating set S in a graph G is called independent injective dominating set (IInj-d... more An injective dominating set S in a graph G is called independent injective dominating set (IInj-dominating set) if for every v ∈ S, N (v) ∩ S = φ. The minimum cardinality of such injective dominating set is called independent injective domination number of G and denoted by γ i i n (G). In this paper, we introduce the independent injective domination of graphs and we define the IID-graph. Exact values for some families of graphs, relations with the other domination parameters are obtained. Also, we introduce the independent injective frustration number of graphs. Bounds and some interesting results are established.
Vn-Arithmetic graph has been introduced by Vasumathi and
Vangipuram [9]. In this paper some prope... more Vn-Arithmetic graph has been introduced by Vasumathi and Vangipuram [9]. In this paper some properties of Vn-rithmetic graph, maximum degree, minimum degree, number of edges, diameter,radius, Hamiltonian and Eulerian are studied. Also, we introduce m-Arithmetical graphs. Some properties and interesting results for m-Arithmetical graphs are established.
In this paper, we introduce a new type of graph energy called the non-common-neighborhood energy ... more In this paper, we introduce a new type of graph energy called the non-common-neighborhood energy E_{NCN}(G). NCN-energy for some standard graphs is obtained and an upper bound for E_{NCN}(G) is found when G is a strongly regular graph. Also the relation between common neighbourhood energy and non-common neighbourhood energy of a graph is established.
Let G = (V, E) be a simple graph. An injective dominating set S in G is called injective stable d... more Let G = (V, E) be a simple graph. An injective dominating set S in G is called injective stable dominating set (Inj-stable dominating set) if for every vertex v ∈ S, N in (v) ∩ S = φ. The minimum cardinality of such injective dominating set is called injective stable domination number (Inj-stable domination number) of G and denoted by i in (G). In this research work, we introduce the injective stable domination in graphs. Exact values for some families of graphs, relations with the other domination parameters are obtained. Bounds and some interesting results are established.
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Papers by Anwar Alwardi
Vangipuram [9]. In this paper some properties of Vn-rithmetic
graph, maximum degree, minimum degree, number of edges, diameter,radius, Hamiltonian and Eulerian are studied. Also, we introduce m-Arithmetical graphs. Some properties and interesting results for m-Arithmetical graphs are established.
Vangipuram [9]. In this paper some properties of Vn-rithmetic
graph, maximum degree, minimum degree, number of edges, diameter,radius, Hamiltonian and Eulerian are studied. Also, we introduce m-Arithmetical graphs. Some properties and interesting results for m-Arithmetical graphs are established.