The divisibility graph D (X) is a directed graph with vertex set X\{1} and an arc from a to b whenever a divides b. Since the divisibility graph and its underlying graph have the same number of connected components, we consider the underlying graph of D (X), and we denote it by D(X). In this paper, we will find some graph theoretical parameters of D(X), some relations between the structure of D(X), and the structure of known graphs Γ(X), ∆(X) and B(X) will be considered. In addition, we investigate some properties of D(XY ) for the product of two non-empty subsets X and Y of positive integers. 2010 AMS Mathematics subject classification. Primary 05C25.

The harmonic index of a graph G ( H G ) is defined as the sum of the weights 2 / d u + d v for all edges u v of G , where d u is the degree of a vertex u in G . In this paper, we show that H G ≥ D G + 5 / 3 − n / 2 and H G ≥ 1 / 2 + 2 / 3 n − 2 D G , where G is a quasi-tree graph of order n and diameter D G . Indeed, we show that both lower bounds are tight and identify all quasi-tree graphs reaching these two lower bounds.

Abstract. Let X be a non-empty set of positive integers and X * = X \ {1}.The divisibility graph D(X) has X * as the vertex set and there is an edge connecting a and b with a, b ∈ X * whenever a divides b or b divides a. Let

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The harmonic index of a graph G, is defined as the sum of weights 2/d(u)+d(v) of all edges uv of G, where d(u) is the degree of the vertex u in G. In this paper we find the minimum harmonic index of bicyclic graph of order n and diameter d. We also characterized all bicyclic graphs reaching the minimum bound.