We design an O(m) algorithm to find a minimum weighted colouring and a maximum weighted clique of a perfectly ordered graph. We also present two O(n’) algorithms to find a minimum weighted colouring of a comparability graph and of a triangulated graph. Our colouring algorithms use an algorithm to find a stable set meeting all maximal (with respect to set inclusion) cliques of a perfectly ordere...

The energy of a graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a graph G is equal to the sum of the absolute value of the difference between the eigenvalues of the Laplacian matrix of G and the average degree of the vertices of G. Motivated by the work from Sharafdini an...

In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weighted K2 (K2)bundles over a weighted graph F can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs are F As an application, we compute the signature of a link whose corresponding weighted...

The purpose of the matching interdiction problem in a weighted graph is to find two vertices such that the weight of the maximum matching in the graph without these vertices is minimized. An approximate solution for this problem has been presented. In this paper, we consider dendrimers as graphs such that the weights of edges are the bond lengths. We obtain the maximum matching in some types of...

the purpose of the matching interdiction problem in a weighted graph is to findtwo vertices such that the weight of the maximum matching in the graph withoutthese vertices is minimized. an approximate solution for this problem has beenpresented. in this paper, we consider dendrimers as graphs such that the weightsof edges are the bond lengths. we obtain the maximum matching in some types ofdend...

A graph is called a weighted graph when each 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 with v. For a subgraph H of a weighted graph G, the weight of H is the sum of the weights of the edges belonging to H . In this paper, we give a new sufficient condition for a weighted graph to...

We introduce a generalization of competition graphs, called weighted competition graphs. The weighted competition graph of a digraph D = (V,A), denoted by Cw(D), is an edge-weighted graph (G,w) such that G = (V,E) is the competition graph of D, and the weight w(e) of an edge e = xy ∈ E is the number of the common preys of x and y in D. We investigate properties of weighted competition graphs.

This paper proposes an algorithm for buUdmg weighted directed graph,defmes the weighted directed relationship matrix of the graph,an d describes algorithm implementation using this matrix.Based on this algorithm ,an effective way for building an d drawing weighted directed graphs is presented,forming a foundation for visual implementation of the algorithm in the graph theory.