Let G(V, E) be a Graph. The common neighborhood Graph (conGraph) of G is a Graph with vertex set V, in which two vertices are adjacent if and only if they have a common neighbor in G. In this paper, we obtain characteristics of conGraphs under Graph operations; Graph union, Graph cartesian product, Graph tensor product, and Graph join, and relations between Cayley Graphs and its conGraphs.

Background and Aim: Different surgical techniques have been proposed for crown exposure among patients under orthodontic treatment. The aim of this study was to investigate periodontal indices in closed flap operation technique used for impacted teeth crown exposure.Materials and Methods: In this descriptive study (2001-2003) 20 patients (15 females and 5 males), age ranging from 15 to 20 years, were participated. Following the performance of closed flap operation technique, in 12 patients impacted canine and central incisors were exposed and treated orthodontically. Periodontal parameters including sulcular depth, width of keratinized gingiva and attachment loss were recorded. Results: The mean of gingival sulcus depth was 1.50±0.43, the means of keratinized gingival in central incisors and canines were 3.07±0.53 and 2.30±0.42, respectively. In one patient, attachment loss was reported.Conclusion: Considering the optimal sulcular depth and keratinized gingiva, the closed flap operation technique seems suitable.

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In this paper, we first collect the earlier results about some Graph operations and then we present applications of these results in working with chemical Graphs.

Distance-balanced Graphs are introduced as Graphs in which every edge uv has the following property: the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. Basic properties of these Graphs are obtained. In this paper, we study the conditions under which some Graph operations produce a distance-balanced Graph.

The distance d (u, v) between two vertices u and v of a Graph Gis equal to the length of a shortest path that connects u and v. Define WW (G, x) =1/2 S [a, b]ÍV (G) x d (a, b)+d2 (a, b), where d (G) is the greatest distance between any two vertices. In this paper the hyper-Wiener polynomials of the Cartesian product, composition, join and disjunction of Graphs are computed.

The independence Graph Ind(G) of a Graph G is the Graph with vertices as maximum independent sets of G and two vertices are adjacent, if and only if the corresponding maximum independent sets are disjoint. In this work, we find the independence Graph of Cartesian product of d copies of complete Graphs Kq, which is known as the Hamming Graph H(d, q). Greenwell and Lovasz [7] found that the independence number of direct product of d copies of Kq as qd−1. We prove that the independence number of Hamming Graph H(d, q), which is cartesian product of d copies of Kq, is also qd−1. As an application of our findings, we find answers for rook problem in higher dimensional square chess board.

In this paper, we find the star chromatic number of central Graph of complete bipartite Graph and corona Graph of complete Graph with path and cycle.

Let G be a simple Graph with vertex set {n1, n2, … , nn}. The common neighborhood Graph of G, denoted by con (G), is a Graph with vertex set {n1, n2, … , nn}, in which two vertices are adjacent if and only if they have at least one common neighbor in the Graph G. In this paper, we compute the common neighborhood of some composite Graphs. In continue, we investigate the relation between hamiltonicity of Graph G and con (G). Also, we obtain a lower bound for the clique number of con (G) in terms of clique number of Graph G. Finally we state that the total chromatic number of G is bounded by chromatic number of con (T(G)).