In this paper, we obtain the domination polynomial of some coalescence of a complete Kn and wheel Wn graphs such as edge introducing, vertex identification, coneighboracation graph, edge identified graph, t-tuple coalescence graphs and b-bridge tuple coalescence graphs with some special graphs.

In this paper, we define a new concept in theory of graph energy called coneighbor energy of a graph G, denoted by CE(G), as the sum of the absolute value of eigenvalues of its coneighbor matrix. We are going to derive some results about the coneighbor energy of graphs and coneighboracation graph that possess some coneighbor vertices.

In this paper, we introduce new results of vertex connected dominating set and vertex connected domination polynomial of vertex identification, edge introduced and t-tuple of complete graph, also we determine new results of vertex connected dominating set and vertex connected domination polynomial of vertex identification, edge introduced and t-tuple of wheel graph .

Let G = (V, E) be a simple connected undirected graph. In this paper, we define generalized Euler's Φ-Function of a graph which is the summation of the Euler's ϕ−function of the degree of the vertices of a graph and it is denoted by Φ (G). It is determined the general form of Euler's Φ-function of some standard graphs. Finally, some important results and properties are studied.

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