Bounds for the Largest Laplacian Eigenvalue of Weighted Graphs
نویسنده
چکیده
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 graph with each of its edges bearing a positive definite matrix as weight,” unless otherwise stated. The notations to be used in paper are given in the following. Let G be a weighted graph on n vertices. Denote by w i,j the positive definite weight matrix of order p of the edge ij, and assume that w ij = w ji . We write i ∼ j if vertices i and j are adjacent. Let w i = ∑ j:j∼i w ij . be the weight matrix of the vertex i. The Laplacian matrix of a graph G is defined as L(G) =
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