A set D of vertices in a graph G is k-fair dominating if every vertex not adjacent to exactly k D. The weighted domination number $$\mathrm {wfd}_k(G)$$ vertex-weighted the minimum weight w(D) among all sets In addition number, some auxiliary parameters are defined. It shown that for cactus graph, and can be calculated linear time.

Weconsider a directed graph on the integerswith a directed edge fromvertex i to j present with probability n−1, whenever i < j , independently of all other edges. Moreover, to each edge (i, j) we assign weight n−1(j − i). We show that the closure of vertex 0 in such a weighted random graph converges in distribution to the Poisson-weighted infinite tree as n → ∞. In addition, we derive limit the...

We introduce the concept of constant 2-labelling of a vertex-weighted graph and show how it can be used to obtain perfect weighted coverings. Roughly speaking, a constant 2-labelling of a vertex-weighted graph is a black and white colouring of its vertex set which preserves the sum of the weights of black vertices under some automorphisms. We study constant 2-labellings on four types of vertex-...

Aspect-based sentiment analysis is a fine-grained that focuses on the polarity of different aspects text, and most current research methods use combination dependent syntactic graphical neural networks. In this paper, graph attention network aspect-based model based weighting dependencies (WGAT) designed to address problem in traditional models do not sufficiently analyse types dependencies; pr...

A weighted graph is a graph in which each edge e is assigned a non-negative number w(e), called the weight of e. In this paper, some theorems on the existence of long paths and cycles in unweighted graphs are generalized to heavy paths and cycles in weighted graphs.

A graph is distance-hereditary if the distance between any two vertices in a connected induced subgraph is the same as in the original graph. This paper presents e-cient algorithms for solving the weighted connected k-domination and the weighted k-dominating clique problems in distance-hereditary graphs. c © 2001 Elsevier Science B.V. All rights reserved.

We prove that for every d > 1 and every undirected, weighted graph G = (V, E), there exists a weighted graph H with at most ⌈d |V |⌉ edges such that for every x ∈ IR , 1 ≤ x T LHx x LGx ≤ d + 1 + 2 √ d d + 1 − 2 √ d , where LG and LH are the Laplacian matrices of G and H , respectively.