As the extension of the previous work by Ciucu and the present authors (J. Combin. Theory Ser. A 112(2005) 105–116), this paper considers the problem of enumeration of spanning trees of weighted graphs with an involution which allows fixed points. We show that if G is a weighted graph with an involution, then the sum of weights of spanning trees of G can be expressed in terms of the product of ...

Let G = (V, E) be simple graphs, as graphs which have no loops or parallel edges such that V is a finite set of vertices and E is a set of edges. A weighted graph is a graph each edge of which has been assigned to a square matrix called the weight of the edge. All the weightmatrices are assumed to be of same order and to be positive matrix. In this paper, by “weighted graph” we mean “a weighted...

The Maximum Weighted Independent Set (MWIS) problem is a classic graph optimization NP-hard problem. Given an undirected graph G = (V, E) and weighting function defined on the vertex set, the MWIS problem is to find a vertex set S V whose total weight is maximum subject to no two vertices in S are adjacent. This paper presents a novel approach to approximate the MWIS of a graph using minimum we...

An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the vertex set of the graph. The minimum weight efficient domination problem is the problem of finding an efficient dominating set of minimum weight in a given vertex-weighted graph; the maximum weight efficient domination problem is defined similarly. We develop a framework...

Spectral graph theory has proven to be a useful tool in the analysis of high-dimensional data sets. Recall that, mathematically, a graph is a collection of objects (nodes) and connections between them (edges); a weighted graph additionally assigns numerical values (weights) to the edges. Graphs are represented by their adjacency whose elements are the weights between the nodes. Spectral graph t...

In this paper, a new authorization delegation model is proposed, which is based on weighted directed graph to present the trust relationship between the entities and the existing issues, such as the evaluation of the trust and the presentation of the trust among the entities are addressed in simple and visual way. The model and the corresponding algorithm of the compliance checking is analyzed ...

We study the graph clustering problem where each observation (edge or no-edge between a pair of nodes) may have a different level of confidence/uncertainty. We propose a clustering algorithm that is based on optimizing an appropriate weighted objective, where larger weights are given to observations with lower uncertainty. Our approach leads to a convex optimization problem that is efficiently ...

We consider a weighted Cheeger’s constant for a graph and we examine the gap between the first two eigenvalues of Laplacian. We establish several isoperimetric inequalities concerning the unweighted Cheeger’s constant, weighted Cheeger’s constants and eigenvalues for Neumann and Dirichlet conditions .

Graph partitioning is a technique which is widely used in many fields of computer science and engineering. The goal is to partition the vertices of a graph into a certain number of disjoint sets of approximately the same size, so that a cut metric is minimized. Due to the NP-hardness of the problem and its practical importance, many different heuristics (spectral, combinatorial, evolutionist, e...