Search

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]