scholarly journals On the Spanning Trees of the Hypercube and other Products of Graphs

10.37236/2510 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Olivier Bernardi
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Cartesian Product ◽  
Product Formula ◽  
General Context ◽  
Complete Graphs ◽  
Independence Property ◽  
Number Of Spanning Trees ◽  
Tree Approach

We give two combinatorial proofs of a product formula for the number of spanning trees of the $n$-dimensional hypercube. The first proof is based on the assertion that if one chooses a uniformly random rooted spanning tree of the hypercube and orient each edge from parent to child, then the parallel edges of the hypercube get orientations which are independent of one another. This independence property actually holds in a more general context and has intriguing consequences. The second proof uses some "killing involutions'' in order to identify the factors in the product formula. It leads to an enumerative formula for the spanning trees of the $n$-dimensional hypercube augmented with diagonals edges, counted according to the number of edges of each type. We also discuss more general formulas, obtained using a matrix-tree approach, for the number of spanning trees of the Cartesian product of complete graphs.

scholarly journals A FORMULA FOR THE NUMBER OF SPANNING TREES IN CIRCULANT GRAPHS WITH NONFIXED GENERATORS AND DISCRETE TORI

2015 ◽  
Vol 92 (3) ◽  
pp. 365-373 ◽  
Author(s):  
JUSTINE LOUIS
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Closed Formula ◽  
Circulant Graphs ◽  
The Matrix ◽  
Number Of Spanning Trees

We consider the number of spanning trees in circulant graphs of ${\it\beta}n$ vertices with generators depending linearly on $n$. The matrix tree theorem gives a closed formula of ${\it\beta}n$ factors, while we derive a formula of ${\it\beta}-1$ factors. We also derive a formula for the number of spanning trees in discrete tori. Finally, we compare the spanning tree entropy of circulant graphs with fixed and nonfixed generators.

scholarly journals Number of Spanning Trees of Different Products of Complete and Complete Bipartite Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-23 ◽  
Author(s):  
S. N. Daoud
Keyword(s):  
Tensor Product ◽  
Linear Algebra ◽  
Spanning Trees ◽  
Matrix Theory ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Normal Product ◽  
Complete Bipartite Graphs ◽  
Composition Product ◽  
Number Of Spanning Trees

Spanning trees have been found to be structures of paramount importance in both theoretical and practical problems. In this paper we derive new formulas for the complexity, number of spanning trees, of some products of complete and complete bipartite graphs such as Cartesian product, normal product, composition product, tensor product, symmetric product, and strong sum, using linear algebra and matrix theory techniques.

scholarly journals Spanning trees in random regular uniform hypergraphs

2021 ◽  
pp. 1-25
Author(s):  
Catherine Greenhill ◽  
Mikhail Isaev ◽  
Gary Liang
Keyword(s):  
Asymptotic Distribution ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Threshold Value ◽  
Regular Graphs ◽  
Cubic Graphs ◽  
Trivial Case ◽  
Uniform Hypergraphs ◽  
Vertex Set ◽  
Number Of Spanning Trees

Abstract Let $${{\mathcal G}_{n,r,s}}$$ denote a uniformly random r-regular s-uniform hypergraph on the vertex set {1, 2, … , n}. We establish a threshold result for the existence of a spanning tree in $${{\mathcal G}_{n,r,s}}$$ , restricting to n satisfying the necessary divisibility conditions. Specifically, we show that when s ≥ 5, there is a positive constant ρ(s) such that for any r ≥ 2, the probability that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree tends to 1 if r > ρ(s), and otherwise this probability tends to zero. The threshold value ρ(s) grows exponentially with s. As $${{\mathcal G}_{n,r,s}}$$ is connected with probability that tends to 1, this implies that when r ≤ ρ(s), most r-regular s-uniform hypergraphs are connected but have no spanning tree. When s = 3, 4 we prove that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree with probability that tends to 1, for any r ≥ 2. Our proof also provides the asymptotic distribution of the number of spanning trees in $${{\mathcal G}_{n,r,s}}$$ for all fixed integers r, s ≥ 2. Previously, this asymptotic distribution was only known in the trivial case of 2-regular graphs, or for cubic graphs.

scholarly journals Listing all spanning trees in Halin graphs — sequential and Parallel view

2018 ◽  
Vol 10 (01) ◽  
pp. 1850005
Author(s):  
K. Krishna Mohan Reddy ◽  
P. Renjith ◽  
N. Sadagopan
Keyword(s):  
Parallel Algorithm ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Combinatorial Problem ◽  
Halin Graph ◽  
Labeled Graph ◽  
Number Of Leaves ◽  
Acyclic Subgraph ◽  
Number Of Spanning Trees ◽  
Halin Graphs

For a connected labeled graph [Formula: see text], a spanning tree [Formula: see text] is a connected and acyclic subgraph that spans all vertices of [Formula: see text]. In this paper, we consider a classical combinatorial problem which is to list all spanning trees of [Formula: see text]. A Halin graph is a graph obtained from a tree with no degree two vertices and by joining all leaves with a cycle. We present a sequential and parallel algorithm to enumerate all spanning trees in Halin graphs. Our approach enumerates without repetitions and we make use of [Formula: see text] processors for parallel algorithmics, where [Formula: see text] and [Formula: see text] are the depth, the number of leaves, respectively, of the Halin graph. We also prove that the number of spanning trees in Halin graphs is [Formula: see text].

scholarly journals The Evaluation of the Number and the Entropy of Spanning Trees on Generalized Small-World Networks

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Raihana Mokhlissi ◽  
Dounia Lotfi ◽  
Joyati Debnath ◽  
Mohamed El Marraki ◽  
Noussaima EL Khattabi
Keyword(s):  
Complex Networks ◽  
Spanning Tree ◽  
Decomposition Method ◽  
Spanning Trees ◽  
Small World ◽  
Theoretical Computer Science ◽  
The Other ◽  
Small World Networks ◽  
Theoretical Computer ◽  
Number Of Spanning Trees

Spanning trees have been widely investigated in many aspects of mathematics: theoretical computer science, combinatorics, so on. An important issue is to compute the number of these spanning trees. This number remains a challenge, particularly for large and complex networks. As a model of complex networks, we study two families of generalized small-world networks, namely, the Small-World Exponential and the Koch networks, by changing the size and the dimension of the cyclic subgraphs. We introduce their construction and their structural properties which are built in an iterative way. We propose a decomposition method for counting their number of spanning trees and we obtain the exact formulas, which are then verified by numerical simulations. From this number, we find their spanning tree entropy, which is lower than that of the other networks having the same average degree. This entropy allows quantifying the robustness of the networks and characterizing their structures.

scholarly journals Parallel strategies for a multi-criteria GRASP algorithm

Production ◽  
2007 ◽  
Vol 17 (1) ◽  
pp. 84-93 ◽  
Author(s):  
Dalessandro Soares Vianna ◽  
José Elias Claudio Arroyo ◽  
Pedro Sampaio Vieira ◽  
Thiago Ribeiro de Azeredo
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Efficient Solutions ◽  
Search Problem ◽  
Pareto Optimal ◽  
Complete Graphs ◽  
Pareto Optimal Solutions ◽  
Adaptive Search ◽  
Optimal Solutions

This paper proposes different strategies of parallelizing a multi-criteria GRASP (Greedy Randomized Adaptive Search Problem) algorithm. The parallel GRASP algorithm is applied to the multi-criteria minimum spanning tree problem, which is NP-hard. In this problem, a vector of costs is defined for each edge of the graph and the goal is to find all the efficient or Pareto optimal spanning trees (Pareto-optimal solutions). Each process finds a subset of efficient solutions. These subsets are joined using different strategies to obtain the final set of efficient solutions. The multi-criteria GRASP algorithm with the different parallel strategies are tested on complete graphs with n = 20, 30 and 50 nodes and r = 2 and 3 criteria. The computational results show that the proposed parallel algorithms reduce the execution time and the results obtained by the sequential version were improved.

scholarly journals The Number of Spanning Trees of the Cartesian Product of Regular Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Mei-Hui Wu ◽  
Long-Yeu Chung
Keyword(s):  
Simple Formula ◽  
Spanning Trees ◽  
Cartesian Product ◽  
Regular Graphs ◽  
Electrical Circuits ◽  
Important Measure ◽  
Computational Perspective ◽  
Regular Networks ◽  
Number Of Spanning Trees

The number of spanning trees in graphs or in networks is an important issue. The evaluation of this number not only is interesting from a mathematical (computational) perspective but also is an important measure of reliability of a network or designing electrical circuits. In this paper, a simple formula for the number of spanning trees of the Cartesian product of two regular graphs is investigated. Using this formula, the number of spanning trees of the four well-known regular networks can be simply taken into evaluation.

On Spanning Trees in Catacondensed Molecules

1993 ◽  
Vol 48 (10) ◽  
pp. 1026-1030 ◽  
Author(s):  
I. Gutman ◽  
R. B. Mallion
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Number Of Spanning Trees

Abstract Among all catacondensed benzenoid molecules the unbranched systems possess the largest number of spanning trees. Some other relations for the spanning-tree count of catacondensed benzenoids are also reported. The results obtained continue to hold for catacondensed molecules composed of rings the size of which differ from six. The propositions reported do, however, depend for their validity on there being rings of only one size in the catacondensed system under study.

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