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A graph associated to a finite group is a way to analyze some properties of a group graphically. Many graphs of groups have been constructed according to the properties of the groups such as the commuting and non-commuting graphs. Besides, a Roman dominating function (in brief RDF) of a graph Γ with vertex set V (Γ) and edge set E(Γ), defined as a function f from the set V (Γ) to the set of numbers {0, 1, 2} in which for any vertex u ∈ V(Γ) with f(u) = 0 there is at least a vertex v ∈ V(Γ) with f(v) = 2 is adjacent to u. The summation of the values f(u) for all the vertices of Γ is defined by the weight of the RDF f. Meanwhile, the Roman domination number (in short RDN) of Γ, γR(Γ), is the minimum weight of an RDF defined on on Γ [1]. Based on that, the Roman domination polynomial (RDP) of a graph Γ on p vertices is defined by R(Γ, x) = ... r(Γ, j)xj, where r(Γ, j) is the number of RDFs of Γ with weight j [2]. In this paper, the RDPs of commuting and non-commuting graphs associated to dihedral groups with order 2n are computed and some examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

The Harary index H(G) of a connected graph G is defined as the sum of reciprocals of distances between all pairs of vertices in G [9, 13]. In this paper, as a continuing the study of entire indices like entire Zagreb indices of graphs [1], we introduce the entire Harary index by adding the reciprocals of distances between all pairs of edges and all pairs of edges and vertices to the Harary index. Our motivation in doing this coming from the following fact about molecular graphs: The intermolecular forces do not exist only between the atoms, but also between the atoms and bonds, so the relations between the atoms and bonds and also between the bonds together should be taken in account. Exact values of some families of graphs and some graphs operations are obtained. Simple formula of the entire Harary index of trees and some bounds are established. [ABSTRACT FROM AUTHOR]

Topological indices are graph invariants computed usually by means of the distances or degrees of vertices of a graph. In chemical graph theory, a molecule can be modeled by a graph by replacing atoms by the vertices and bonds by the edges of this graph. Topological graph indices have been successfully used in determining the structural properties and in predicting certain physicochemical properties of chemical compounds. Wiener index is the oldest topological index which can be used for analyzing intrinsic properties of a molecular structure in chemistry. The Wiener index of a graph G is equal to the sum of distances between all pairs of vertices of G. Recently, the entire versions of several indices have been introduced and studied due to their applications. Here we introduce the entire Wiener index of a graph. Exact values of this index for trees and some graph families are obtained, some properties and bounds for the entire Wiener index are established. Exact values of this new index for subdivision and k-subdivision graphs and some graph operations are obtained. [ABSTRACT FROM AUTHOR]

The injective domination polynomial of a graph G of order p is defined as Din(G, x) = ∑p j=γin(G) din(G, j)xj ; where γin(G) is the injective domination number of G and din(G, j) is the number of injective dominating sets of G of size j, [3]. We call the roots of an injective domination polynomial of a graph the injective domination roots of that graph. In this article, we compute the injective domination polynomial of all graphs of order less than or equal six and their roots and present them in tables. [ABSTRACT FROM AUTHOR]

A 2-rainbow dominating function (2RDF) g : V → P(A) ( where P(A) is the power set of the set of two colors A = {1, 2} ) of a graph G = (V,E) is defined to be satisfying the condition that for every vertex v ∊ V with g(v) = ϕ we have S u∊ℕ(v) g(u) = A. The minimum value of w(g) = Σ v∊V j∣(v)∣ among all such functions g of G is called the 2-rainbow domination number of G and is denoted by γr2 (G). A set S ⊆ V is a global dominating set of a graph G if S dominates both G and its complement G. The minimum cardinality γg(G) of a global dominating set of G is called the global domination number of the graph G. In this paper, we introduce the global 2-rainbow domination number γgr2 (G) of a graph G, study some of its properties, determine its exact values for some specific graphs and we characterize the graphs G with γgr2 (G) = p, where p is the number of vertices of G. [ABSTRACT FROM AUTHOR]

In this paper, we continue the study of Vn-Arithmetic graph by determining domination parameters of Vn. Bounds for some Vn graph parameters are established. [ABSTRACT FROM AUTHOR]

A subset S of vertices in a graph G is called injective dominating set if for every vertex v not in 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. The injective domination number γin(G) of G is the minimum cardinality of such dominating sets. In this article, we introduce the injective domination polynomial of a graph G of order p as Din(G; x) =∑j=γin(9(G)din(G; j)xj, where din(G; j) is the number of the injective dominating sets of G of size j. We obtain some properties of Din(G; x) and compute this polynomial for some specific graphs. [ABSTRACT FROM AUTHOR]

Let G = (V;E) be a simple graph. A subset D of V is called injective dominating set (Inj-dominating set) if for every vertex v ∊ V - D there exists a vertex u ∊ D such that |Γ(u; v)| ≥ 1, where |Γ(u; v)| is the number of common neighbors between the vertices u and v. The minimum cardinality of such dominating set denoted by γin(G) and is called injective domination number (Inj-domination number) of G. In this paper, we introduce the injective domination of a graph G and analogous to that, we define the injective independence number (Inj-independence number) βin(G) and injective domatic number (Inj-domatic number) din(G). Bounds and some interesting results are established. [ABSTRACT FROM AUTHOR]

Let G = (V,E) be a connected graph. The distance eccentricity neighborhood of u ∈ V (G) denoted by NDe(u) is defined as NDe(u) = {v ∈ V (G) : d(u, v) = e(u)}, where e(u) is the eccentricity of u. The cardinality of NDe(u) is called the distance eccentricity degree of the vertex u in G and denoted by degDe(u). In this paper, we introduce the first and second distance eccentricity Zagreb indices of a connected graph G as the sum of the squares of the distance eccentricity degrees of the vertices, and the sum of the products of the distance eccentricity degrees of pairs of adjacent vertices, respectively. Exact values for some families of graphs and graph operations are obtained. [ABSTRACT FROM AUTHOR]

The Roman domination polynomial of a graph G of order p is defined as R(G, x) = ∑2n j=γR(G) r(G, j)x j, where r(G, j) is the number of Roman dominating functions of G of weight j [5]. The roots of a Roman domination polynomial of a graph are called the Roman domination roots of that graph. In this article, the Roman domination polynomials of all the connected graphs of order less than or equal to six are obtained and their roots are computed. Furthermore, all these graphs and their Roman domination polynomials and roots are illustrated in a table. [ABSTRACT FROM AUTHOR]