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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]

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]