scholarly journals Product distance matrix of a graph and squared distance matrix of a tree

2013 ◽  
Vol 7 (2) ◽  
pp. 285-301 ◽  
Author(s):  
R.B. Bapat ◽  
S. Sivasubramanian
Keyword(s):  
Directed Graph ◽  
Distance Matrix ◽  
Directed Tree ◽  
Edge Orientation ◽  
Weighted Directed Graph ◽  
Orientation Matrix ◽  
Strongly Connected

Let G be a strongly connected, weighted directed graph. We define a product distance ?(i,j) for pairs i,j of vertices and form the corresponding product distance matrix. We obtain a formula for the determinant and the inverse of the product distance matrix. The edge orientation matrix of a directed tree is defined and a formula for its determinant and its inverse, when it exists, is obtained. A formula for the determinant of the (entry-wise) squared distance matrix of a tree is proved.

scholarly journals A Combinatorial Proof of a Formula of Biane and Chapuy

10.37236/7061 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Sinho Chewi ◽  
Venkat Anantharam
Keyword(s):  
Directed Graph ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Outgoing Edge ◽  
Alternative Proof ◽  
Combinatorial Proof ◽  
Tree Graph ◽  
Weighted Directed Graph ◽  
Strongly Connected

Let $G$ be a simple strongly connected weighted directed graph. Let $\mathcal{G}$ denote the spanning tree graph of $G$. That is, the vertices of $\mathcal{G}$ consist of the directed rooted spanning trees on $G$, and the edges of $\mathcal{G}$ consist of pairs of trees $(t_i, t_j)$ such that $t_j$ can be obtained from $t_i$ by adding the edge from the root of $t_i$ to the root of $t_j$ and deleting the outgoing edge from $t_j$. A formula for the ratio of the sum of the weights of the directed rooted spanning trees on $\mathcal{G}$ to the sum of the weights of the directed rooted spanning trees on $G$ was recently given by Biane and Chapuy. Our main contribution is an alternative proof of this formula, which is both simple and combinatorial.

The Algebraic Algorithm for Solving the Path Planning Problem of a Weighted Directed Graph

2014 ◽  
Vol 644-650 ◽  
pp. 1648-1653 ◽  
Author(s):  
Yan Ping Li ◽  
Kun Wei ◽  
Dan Wang
Keyword(s):  
Path Planning ◽  
Directed Graph ◽  
Shortest Path ◽  
Distance Matrix ◽  
Experimental Results ◽  
Planning Problem ◽  
Dijkstra Algorithm ◽  
Weighted Directed Graph ◽  
Direct Distance ◽  
Path Planning Problem

In this paper, an algebraic algorithm is developed with Min-algebra for the path planning problem of a simple weighted directed graph. According to the algebraic algorithm, the shortest path and its minimum steps will be concluded through the direct distance matrix A. Experimental results show that the algebraic algorithm is suitable for the path planning problem. The shortest path and its length can be gotten rapidly based on the proposed algorithm. Finally, the results are compared to those obtained by the conventional Dijkstra algorithm.

scholarly journals Convert a Strongly Connected Directed Graph to a Black-and-White 3-SAT Problem by the Balatonboglár Model

2020 ◽  
Author(s):  
Gábor Kusper ◽  
Csaba Biró
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Deep Insight ◽  
Communication Graph ◽  
Sat Problem ◽  
Weak Model ◽  
Know How ◽  
Black And White ◽  
Communication Graphs ◽  
Strongly Connected

In a previous paper we defined the Black-and-White SAT problem which has exactly two solutions, where each variable is either true or false. We showed that Black-and-White $2$-SAT problems represent strongly connected directed graphs. We presented also the strong model of communication graphs. In this work we introduce two new models, the weak model, and the Balatonbogl\'{a}r model of communication graphs. A communication graph is a directed graph, where no self loops are allowed. In this work we show that the weak model of a strongly connected communication graph is a Black-and-White SAT problem. We prove a powerful theorem, the so called Transitions Theorem. This theorem states that for any model which is between the strong and the weak model, we have that this model represents strongly connected communication graphs as Blask-and-White SAT problems. We show that the Balatonbogl\'{a}r model is between the strong and the weak model, and it generates $3$-SAT problems, so the Balatonbogl\'{a}r model represents strongly connected communication graphs as Black-and-White $3$-SAT problems. Our motivation to study these models is the following: The strong model generates a $2$-SAT problem from the input directed graph, so it does not give us a deep insight how to convert a general SAT problem into a directed graph. The weak model generates huge models, because it represents all cycles, even non-simple cycles, of the input directed graph. We need something between them to gain more experience. From the Balatonbogl\'{a}r model we learned that it is enough to have a subset of a clause, which represents a cycle in the weak model, to make the Balatonbogl\'{a}r model more compact. We still do not know how to represent a SAT problem as a directed graph, but this work gives a strong link between two prominent fields of formal methods: SAT problem and directed graphs.

scholarly journals A Note on Hamiltonian Bypasses in Digraphs with Large Degrees

2020 ◽  
pp. 7-17
Author(s):  
Samvel Darbinyan
Keyword(s):  
Directed Graph ◽  
Hamiltonian Path ◽  
Terminal Vertex ◽  
Initial Vertex ◽  
Strongly Connected

Let D be a 2-strongly connected directed graph of order p ≥ 3. Suppose that d(x) ≥ p for every vertex x ∈ V (D) \ {x0}, where x0 is a vertex of D. In this paper, we show that if D is Hamiltonian or d(x0) > 2(p − 1)/5, then D contains a Hamiltonian path, in which the initial vertex dominates the terminal vertex.

scholarly journals COMBINATORIAL TECHNIQUES FOR DETERMINING RANK AND IDEMPOTENT RANK OF CERTAIN FINITE SEMIGROUPS

2002 ◽  
Vol 45 (3) ◽  
pp. 617-630 ◽  
Author(s):  
Inessa Levi ◽  
Steve Seif
Keyword(s):  
Positive Integer ◽  
Directed Graph ◽  
Subject Classification ◽  
Generating Set ◽  
Finite Semigroups ◽  
Idempotent Rank ◽  
Strongly Connected ◽  
Primary 20M20

AbstractLet $\tau$ be a partition of the positive integer $n$. A partition of the set $\{1,2,\dots,n\}$ is said to be of type $\tau$ if the sizes of its classes form the partition $\tau$ of $n$. It is known that the semigroup $S(\tau)$, generated by all the transformations with kernels of type $\tau$, is idempotent generated. When $\tau$ has a unique non-singleton class of size $d$, the difficult Middle Levels Conjecture of combinatorics obstructs the application of known techniques for determining the rank and idempotent rank of $S(\tau)$. We further develop existing techniques, associating with a subset $U$ of the set of all idempotents of $S(\tau)$ with kernels of type $\tau$ a directed graph $D(U)$; the directed graph $D(U)$ is strongly connected if and only if $U$ is a generating set for $S(\tau)$, a result which leads to a proof if the fact that the rank and the idempotent rank of $S(\tau)$ are both equal to$$ \max\biggl\{\binom{n}{d},\binom{n}{d+1}\biggr\}. $$AMS 2000 Mathematics subject classification: Primary 20M20; 05A18; 05A17; 05C20

Strongly Connected Spanning Subgraph for Almost Symmetric Networks

2017 ◽  
Vol 27 (03) ◽  
pp. 207-219
Author(s):  
A. Karim Abu-Affash ◽  
Paz Carmi ◽  
Anat Parush Tzur
Keyword(s):  
Directed Graph ◽  
Euclidean Distance ◽  
Directed Graphs ◽  
Minimum Weight ◽  
Shortest Paths ◽  
Strong Connectivity ◽  
Spanning Subgraph ◽  
Strongly Connected ◽  
Set Of Points ◽  
Symmetric Networks

In the strongly connected spanning subgraph ([Formula: see text]) problem, the goal is to find a minimum weight spanning subgraph of a strongly connected directed graph that maintains the strong connectivity. In this paper, we consider the [Formula: see text] problem for two families of geometric directed graphs; [Formula: see text]-spanners and symmetric disk graphs. Given a constant [Formula: see text], a directed graph [Formula: see text] is a [Formula: see text]-spanner of a set of points [Formula: see text] if, for every two points [Formula: see text] and [Formula: see text] in [Formula: see text], there exists a directed path from [Formula: see text] to [Formula: see text] in [Formula: see text] of length at most [Formula: see text], where [Formula: see text] is the Euclidean distance between [Formula: see text] and [Formula: see text]. Given a set [Formula: see text] of points in the plane such that each point [Formula: see text] has a radius [Formula: see text], the symmetric disk graph of [Formula: see text] is a directed graph [Formula: see text], such that [Formula: see text]. Thus, if there exists a directed edge [Formula: see text], then [Formula: see text] exists as well. We present [Formula: see text] and [Formula: see text] approximation algorithms for the [Formula: see text] problem for [Formula: see text]-spanners and for symmetric disk graphs, respectively. Actually, our approach achieves a [Formula: see text]-approximation algorithm for all directed graphs satisfying the property that, for every two nodes [Formula: see text] and [Formula: see text], the ratio between the shortest paths, from [Formula: see text] to [Formula: see text] and from [Formula: see text] to [Formula: see text] in the graph, is at most [Formula: see text].

An Approach of Process Planning Using Ant Colony Optimization Algorithm

2014 ◽  
Vol 978 ◽  
pp. 209-212
Author(s):  
Jin Feng Wang ◽  
Kai Yu Chu ◽  
Qing Yu Wang
Keyword(s):  
Ant Colony Optimization ◽  
Directed Graph ◽  
Process Planning ◽  
Ant Colony ◽  
Production Costs ◽  
Planning Problem ◽  
Total Production ◽  
Ant Colony Optimization Algorithm ◽  
Process Plan ◽  
Weighted Directed Graph

An ant colony optimization (ACO) approach has been developed to deal with process planning problem to achieve the optimal process plan. A disjunctive weighted directed graph is conducted to describe the operations, precedence constraints between operations, and the possible visited path between operation nodes. A represent of process plan is described based on the disjunctive weighted directed graph. Ant colony goes through the necessary nodes on the graph to achieve the optimal solution with the objective of minimizing Total Production Costs (TPC). Extensive comparative experiments have been carried out to demonstrate the feasibility and efficiency of the proposed approach.

The Exponents of Strongly Connected Graphs

1969 ◽  
Vol 21 ◽  
pp. 769-782 ◽  
Author(s):  
Edward A. Bender ◽  
Thomas W. Tucker
Keyword(s):  
Directed Graph ◽  
Connected Graphs ◽  
Strongly Connected ◽  
Elementary Circuit ◽  
Image Position

A directed graphG is a set of vertices V and a subset of V × V called the edges of G. A path in G of length k,is such that (vi, vi+1) is an edge of G for 1 ≦ i ≦ k. A directed graph G is strongly connected if there is a path from every vertex of G to every other vertex. A circuit is a path whose two end vertices are equal. An elementary circuit has no other equal vertices. See (1) for a fuller discussion.Let G be a finite, strongly connected, directed graph (fscdg). The kth power Gk of G is the directed graph with the same vertices as G and edges of the form (i, j) where G has a path of length k from i to j.

Sign in / Sign up

Export Citation Format

Share Document