China
Africa
Explore chapters and articles related to this topic
Discrete Mathematics
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
empty graph: A graph is empty if it has no edges; K¯ $ \bar{K} $ denotes an empty graph of order n.
W polynomials
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
Note that by deleting the end-vertices of an edge, we may end up with a completely empty graph having no vertices. This possibility is dealt with by taking k=0, in the first case.
A value for games with a priori incompatible players
Published in International Journal of General Systems, 2020
J. M. Gallardo, N. Jiménez, A. Jiménez-Losada, E. Lebrón
Let N be a finite set. The complete graph on N, which contains all the feasible edges, will be denoted by , whereas denotes the empty graph on N, which is edgeless. The family of graphs with vertex set N is denoted by . Let . If then the induced subgraph is the graph whose vertex set is S and whose edge set consists of all the edges of g that have both endpoints in S. We say that S is a clique in g if . If then S is said to be independent in g. The family of maximal cliques of g is denoted by and the family of maximal independent sets of g is denoted by . The complement of g is the graph whose vertex set is N and whose edge set consists of the edges which are not present in g. Notice that and .
A practical analysis of sample complexity for structure learning of discrete dynamic Bayesian networks
Published in Optimization, 2022
In this section, the aim is to investigate the effect of the imaginary sample size α, on sample complexity. Several studies conduct that the imaginary sample size has a significant impact on the model discovery of Bayesian Networks. Steck and Jaakkola show that as , the deletion of an edge is more likely to occur in the structure learning of Bayesian Networks [46]. In the same study, they also demonstrate that the number of edges in a network increases when the prior term increases. Silander et al. conduct practical experiments on structure learning to find an optimal α value [47]. They show that learned structure is highly sensitive to the chosen α value. In order to solve this problematic effect of the prior term, they propose a Bayes method for determining the optimal α. Steck provides an analytical approximation to the optimal α value in a predictive sense [48]. The data properties that have the main effect on determining the optimal α value is provided by considering this approximation. Ueno analytically investigates the behaviour of the BDeu metric when and [49,50]. The sensitivity of model discovery to α is investigated, and it is shown that when , BDeu favours an empty graph; and when , BDeu favours a complete graph. In his studies, by considering the issues faced with prior term α, Scutari experimentally and theoretically show that the BDeu score is not accurate when data is sparse, which is the case when the number of samples is less than the appropriate amount [51,52]. He proposes a new scoring method, Bayesian Dirichlet sparse, which is more suitable for sparse data. Because of this significant effect of the imaginary sample size on the model discovery, we also conducted several experiments to see the impact of it on dDBN model discovery.
Reverse-degree-based topological indices of fullerene cage networks
Published in Molecular Physics, 2023
Ali Ahmad, Ali N. A. Koam, Muhammad Azeem
The complement of a graph is obtained by taking the same set of vertices as the original graph, but with edges between any two vertices that are not adjacent in the original graph. For example, the complement of a complete graph is an empty graph, and the complement of an empty graph is a complete graph [13].