Introduction to Graph Theory and Shortest Spanning Trees

2007 ◽  
pp. 105-116
Keyword(s):  
Graph Theory ◽  
Spanning Trees

scholarly journals A Positive Combinatorial Formula for the Complexity of the $q$-Analog of the $n$-Cube

10.37236/2389 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Murali Krishna Srinivasan
Keyword(s):  
Graph Theory ◽  
Explicit Formula ◽  
Classical Result ◽  
Spanning Trees ◽  
Combinatorial Formula ◽  
Algebraic Graph Theory ◽  
The Matrix ◽  
Number Of Spanning Trees

The number of spanning trees of a graph $G$ is called the complexity of $G$. A classical result in algebraic graph theory explicitly diagonalizes the Laplacian of the $n$-cube $C(n)$  and yields, using the Matrix-Tree theorem, an explicit formula for $c(C(n))$. In this paper we explicitly block diagonalize the Laplacian of the $q$-analog $C_q(n)$ of $C(n)$ and use this, along with the Matrix-Tree theorem, to give a positive combinatorial formula for $c(C_q(n))$. We also explain how setting $q=1$ in the formula for $c(C_q(n))$ recovers the formula for $c(C(n))$.

scholarly journals Systemic Risk in Chinese Commodity Futures Markets: A Graph Theory Analysis

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Jinyu Yang
Keyword(s):  
Graph Theory ◽  
Systemic Risk ◽  
Spanning Trees ◽  
Futures Market ◽  
Commodity Futures ◽  
Commodity Futures Markets ◽  
System Risk ◽  
Graph Theory Analysis ◽  
Probable Path ◽  
Most Probable Path

This paper sets out to explore the contagion of systemic risk in Chinese commodity futures market based on specific tools of the graph-theory. More precisely, we use minimum spanning trees as a way to identify the most probable path for the transmission of prices shocks. In the sample of 30 kinds of Chinese commodity futures, we construct the MST and obtain the most probable and the shortest path for the transmission of a prices shock. We find that metal futures play an important role in commodity futures market and copper stands at the heart of the system (The core position of the system is very important for the transmission of system risk). And our results also reveal that when the risk occurs, the MST structure becomes smaller, leading to the most effective transmission path of risk becomes shorter.

Graph Theory for Temporal Structure

2018 ◽  
Author(s):  
Jason Yust
Keyword(s):  
Graph Theory ◽  
Spanning Trees ◽  
Temporal Structure ◽  
Temporal Networks ◽  
Graph Theoretic ◽  
Temporal Structures ◽  
Mathematical Graph

This chapter introduces mathematical graph theory and develops graph-theory concepts that are useful for temporal networks. By generating chord progressions from networks, the potential musical and temporal meaning of graph-theory concepts, especially cycles, is emphasized. A number of concepts related to trees are introduced to show hierarchical aspects of temporal structure, and to allow for a comparison of Fred Lerdahl and Ray Jackendoff’s prolongational trees to temporal structures. This suggests an enrichment of MOPs through spanning trees, and is channelled into a discussion of graph-theoretic algebras, cycle and edge-cut algebras, as they apply to temporal structures.

Trees

2013 ◽  
pp. 48-69
Author(s):  
Seyed Abbas Hosseinijou ◽  
Farhad Kiya ◽  
Hamed Kalantari
Keyword(s):  
Graph Theory ◽  
Spanning Trees ◽  
Search Algorithms ◽  
Rooted Trees ◽  
Tree Search ◽  
Science And Engineering ◽  
Important Type

In this chapter, the authors present a special and important type of graphs named Trees. Although the concept of a tree is simple, many applications of graph theory in science and engineering are related to this type. The topics of this chapter are selected based on theory and application of trees. Rooted trees, spanning trees, fundamental cycles and bounds, and tree search algorithms are covered. However, this chapter mainly focuses on spanning trees and tree search algorithms.

scholarly journals A chip-firing variation and a new proof of Cayley's formula

2013 ◽  
Vol Vol. 15 no. 1 (Graph Theory) ◽  
Author(s):  
Peter Mark Kayll ◽  
Dave Perkins
Keyword(s):  
Graph Theory ◽  
Spanning Trees ◽  
Complete Graphs ◽  
Physical Processes ◽  
Connected Graphs ◽  
Bijective Proof ◽  
Chip Firing ◽  
International Audience ◽  
General Graphs

Graph Theory International audience We introduce a variation of chip-firing games on connected graphs. These 'burn-off' games incorporate the loss of energy that may occur in the physical processes that classical chip-firing games have been used to model. For a graph G=(V,E), a configuration of 'chips' on its nodes is a mapping C:V→ℕ. We study the configurations that can arise in the course of iterating a burn-off game. After characterizing the 'relaxed legal' configurations for general graphs, we enumerate the 'legal' ones for complete graphs Kn. The number of relaxed legal configurations on Kn coincides with the number tn+1 of spanning trees of Kn+1. Since our algorithmic, bijective proof of this fact does not invoke Cayley's Formula for tn, our main results yield secondarily a new proof of this formula.

scholarly journals Spanning trees with a given number of leaves

2018 ◽  
Author(s):  
Thinh D. Nguyen
Keyword(s):  
Graph Theory ◽  
Combinatorial Optimization ◽  
Computer Games ◽  
Spanning Trees ◽  
Computer Game ◽  
Number Of Leaves

We prove the hardness of yet another problem in graph theory with a flavor of computer games, namelyTerra Mystica. Much like works of Demaine and others, we abstract the computer game to have a mathematicalmodel which is in the form of combinatorial optimization. A reduction from 3Sat shows that our claim holds.

Graph Theory, Coding Theory and Block Designs

1975 ◽  
Author(s):  
P. J. Cameron ◽  
J. H. van Lint
Keyword(s):  
Graph Theory ◽  
Coding Theory ◽  
Block Designs
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