The Randić index of a graph G, denoted by R(G), is defined as the sum 1/d(u)d(v) for all edges uv where d(u) denotes degree vertex u in G. In this note, we show that R(L(T))>n4 any tree T order n≥3. A number relevant conjectures are proposed.

A weighted graph is one in which every edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d(v) is the sum of the weights of the edges incident to v. And the weight of a path is the sum of the weights of the edges belonging to it. In this paper, we give a sufficient condition for a weighted graph to have a heavy path which joins two specifie...

Automatic differentiation is a technique for the rule-based transformation of a subprogram that computes some mathematical function into a subprogram that computes the derivatives of that function. Automatic differentiation algorithms are typically expressed as operating on a weighted term graph called a linearized computational graph. Constructing this weighted term graph for imperative progra...

We aim here at obtaining bounds on the first nonzero eigenvalue for selfadjoint boundary value problems on a weighted network by means of equilibrium measures, that include the study of Dirichlet, Neumann and Mixed problems. We also show the sharpness of these bounds throughout the analysis of some examples. In particular we emphasize the case of distance-regular graphs and we show that the obt...

A weighted coloured–edge graph is a graph for which each edge is assigned both a positive weight and a discrete colour, and can be used to model transportation and computer networks in which there are multiple transportation modes. In such a graph paths are compared by their total weight in each colour, resulting in a Pareto set of minimal paths from one vertex to another. This paper will give ...

the dispersibility of graphene is modeled as a mathematical function of the molecular structure of solvent represented by simplified molecular input-line entry systems (smiles) together with the graph of atomic orbitals (gao). the gao is molecular graph where atomic orbitals e.g. 1s1, 2p4, 3d7 etc., are vertexes of the graph instead of the chemical elements used as the graph vertexes in the tra...

We study the application of spectral clustering, prediction and visualization methods to graphs with negatively weighted edges. We show that several characteristic matrices of graphs can be extended to graphs with positively and negatively weighted edges, giving signed spectral clustering methods, signed graph kernels and network visualization methods that apply to signed graphs. In particular,...