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.
Let G = (V,E) be a simple graph. A subset X of E is called edge injective dominating set (edge Inj-dominating set) if for every edge f ∈ E - X there exists an edge g ∈ X such that |Γ(f,g)| ≥ 1, where |Γ(f,g)| is the number of common edge neighbors between the edges f and g. The minimum cardinality of such edge dominating set denoted by γ'in(G) and is called edge injective domination number (edge Inj-domination number) of G. In this paper, we introduce the edge injective domination number, injective independence edge number (Inj-independence edge number) β'in(G) and edge injective domatic number (edge Inj-domatic number) d'in(G) of a graph G. Exact values for some standard graphs, bounds and some interesting results are established.
V n -Arithmetic graph has been introduced by Vasumathi and Vangipuram [9]. In this paper some properties of V n -Arithmetic 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.
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