The Upper Bound of Primitive Exponent of a Class of Special Nonnegative Matrix Pairs

2014 ◽  
Vol 1061-1062 ◽  
pp. 1100-1103
Author(s):  
Mei Jin Luo
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Nonnegative Matrix ◽  
N Cycle ◽  
One To One

There is a one-to-one relationship between nonnegative matrix pairs and two-colored digraph. With the knowledge of graph theory, by studying the associated directed digraph of a class of special nonnegative matrix pairs, that is a class of two-colored digraphs whose uncolored digraph have 3n-1 vertices and consists of one (3n-1)-cycle and one n-cycle are considered. The exponent and characteristic of extreme two-colored digraphs are given.

The Primitive Exponent of a Class of Special Nonnegative Matrix Pairs

2014 ◽  
Vol 915-916 ◽  
pp. 1296-1299
Author(s):  
Mei Jin Luo
Keyword(s):  
Graph Theory ◽  
Nonnegative Matrix ◽  
One To One

There is a one-to-one relationship between nonnegative matrix pairs and two-colored digraph, so the problem of matrices can be changed into that of graphics to be solved. With the knowledge of graph theory, by studying the associated directed digraph of a class of special nonnegative matrix pairs, that is a class of two-colored digraphs whose uncolored digraph have vertices and consists of one-cycle and one-cycle are considered. The exponent and characteristic of extreme two-colored digraphs are given.

scholarly journals An upper bound for the permanent of a nonnegative matrix

1998 ◽  
Vol 281 (1-3) ◽  
pp. 259-263 ◽  
Author(s):  
Suk-Geun Hwang ◽  
Arnold R. Kräuter ◽  
T.S. Michael
Keyword(s):  
Upper Bound ◽  
Nonnegative Matrix

scholarly journals The Diameter of Almost Eulerian Digraphs

10.37236/429 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Peter Dankelmann ◽  
L. Volkmann
Keyword(s):  
Graph Theory ◽  
Directed Graph ◽  
Upper Bound ◽  
Minimum Degree ◽  
Additive Constant ◽  
Undirected Graphs

Soares [J. Graph Theory 1992] showed that the well known upper bound $\frac{3}{\delta+1}n+O(1)$ on the diameter of undirected graphs of order $n$ and minimum degree $\delta$ also holds for digraphs, provided they are eulerian. In this paper we investigate if similar bounds can be given for digraphs that are, in some sense, close to being eulerian. In particular we show that a directed graph of order $n$ and minimum degree $\delta$ whose arc set can be partitioned into $s$ trails, where $s\leq \delta-2$, has diameter at most $3 ( \delta+1 - \frac{s}{3})^{-1}n+O(1)$. If $s$ also divides $\delta-2$, then we show the diameter to be at most $3(\delta+1 - \frac{(\delta-2)s}{3(\delta-2)+s} )^{-1}n+O(1)$. The latter bound is sharp, apart from an additive constant. As a corollary we obtain the sharp upper bound $3( \delta+1 - \frac{\delta-2}{3\delta-5})^{-1} n + O(1)$ on the diameter of digraphs that have an eulerian trail.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

Finding the shortest path for a Hypergraph

2021 ◽  
pp. 2150120
Author(s):  
G. H. Shirdel ◽  
B. Vaez-Zadeh
Keyword(s):  
Graph Theory ◽  
Shortest Path ◽  
Complex Structure ◽  
Dijkstra Algorithm ◽  
Clique Graph ◽  
One To One

A hypergraph is given by [Formula: see text], where [Formula: see text] is a set of vertices and [Formula: see text] is a set of nonempty subsets of [Formula: see text], the member of [Formula: see text] is named hyperedge. So, a hypergraph is a nature generalization of a graph. A hypergraph has a complex structure, thus some researchers try to transform a hypergraph to a graph. In this paper, we define two graphs, Clique graph and Persian graph. These relations are one to one. We can find the shortest path between two vertices in a hypergraph [Formula: see text], by using the Dijkstra algorithm in graph theory on the graphs corresponding to [Formula: see text].

scholarly journals The Study of the Theoretical Size and Node Probability of the Loop Cutset in Bayesian Networks

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1079 ◽  
Author(s):  
Jie Wei ◽  
Yufeng Nie ◽  
Wenxian Xie
Keyword(s):  
Probability Theory ◽  
Graph Theory ◽  
Bayesian Inference ◽  
Bayesian Networks ◽  
Numerical Simulations ◽  
Complete Graph ◽  
Upper Bound ◽  
Numerical Algorithms ◽  
Theoretical Research ◽  
Theoretical Results

Pearl’s conditioning method is one of the basic algorithms of Bayesian inference, and the loop cutset is crucial for the implementation of conditioning. There are many numerical algorithms for solving the loop cutset, but theoretical research on the characteristics of the loop cutset is lacking. In this paper, theoretical insights into the size and node probability of the loop cutset are obtained based on graph theory and probability theory. It is proven that when the loop cutset in a p-complete graph has a size of p − 2 , the upper bound of the size can be determined by the number of nodes. Furthermore, the probability that a node belongs to the loop cutset is proven to be positively correlated with its degree. Numerical simulations show that the application of the theoretical results can facilitate the prediction and verification of the loop cutset problem. This work is helpful in evaluating the performance of Bayesian networks.

scholarly journals A Computational Perspective on Network Coding

2010 ◽  
Vol 2010 ◽  
pp. 1-11
Author(s):  
Qin Guo ◽  
Mingxing Luo ◽  
Lixiang Li ◽  
Yixian Yang
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Lower Bound ◽  
Network Coding ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Multicast Networks ◽  
Computational Perspective ◽  
Source Rate

From the perspectives of graph theory and combinatorics theory we obtain some new upper bounds on the number of encoding nodes, which can characterize the coding complexity of the network coding, both in feasible acyclic and cyclic multicast networks. In contrast to previous work, during our analysis we first investigate the simple multicast network with source rateh=2, and thenh≥2. We find that for feasible acyclic multicast networks our upper bound is exactly the lower bound given by M. Langberg et al. in 2006. So the gap between their lower and upper bounds for feasible acyclic multicast networks does not exist. Based on the new upper bound, we improve the computational complexity given by M. Langberg et al. in 2009. Moreover, these results further support the feasibility of signatures for network coding.

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