Algebraic Graph Theory and Cooperative Control Consensus

2014 ◽  
pp. 23-71 ◽  
Author(s):  
Frank L. Lewis ◽  
Hongwei Zhang ◽  
Kristian Hengster-Movric ◽  
Abhijit Das
Keyword(s):  
Graph Theory ◽  
Cooperative Control ◽  
Algebraic Graph Theory

scholarly journals Order divisor graphs of finite groups

2018 ◽  
Vol 26 (3) ◽  
pp. 29-40
Author(s):  
S. U. Rehman ◽  
A. Q. Baig ◽  
M. Imran ◽  
Z. U. Khan
Keyword(s):  
Graph Theory ◽  
Finite Groups ◽  
Algebraic Graph Theory ◽  
Vertex Set ◽  
Productive Area

AbstractThe interplay between groups and graphs have been the most famous and productive area of algebraic graph theory. In this paper, we introduce and study the graphs whose vertex set is group G such that two distinct vertices a and b having di erent orders are adjacent provided that o(a) divides o(b) or o(b) divides o(a).

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 Computers and discovery in algebraic graph theory

2002 ◽  
Vol 356 (1-3) ◽  
pp. 211-230 ◽  
Author(s):  
Pierre Hansen ◽  
Hadrien Mélot
Keyword(s):  
Graph Theory ◽  
Algebraic Graph Theory

scholarly journals Electrical Networks and Algebraic Graph Theory: Models, Properties, and Applications

2018 ◽  
Vol 106 (5) ◽  
pp. 977-1005 ◽  
Author(s):  
Florian Dorfler ◽  
John W. Simpson-Porco ◽  
Francesco Bullo
Keyword(s):  
Graph Theory ◽  
Electrical Networks ◽  
Algebraic Graph Theory

Consensus for discrete-time heterogeneous networked systems consisting of third-order and first-order agents under fixed and switching topologies

2017 ◽  
Vol 40 (9) ◽  
pp. 2748-2755 ◽  
Author(s):  
Huanyu Zhao ◽  
Shumin Fei
Keyword(s):  
System Theory ◽  
Graph Theory ◽  
Sufficient Condition ◽  
Consensus Problem ◽  
Multi Agent Systems ◽  
Algebraic Graph Theory ◽  
Third Order ◽  
Agent Systems ◽  
First Order ◽  
Multi Agent

This paper investigates the consensus problem for heterogeneous multi-agent systems consisting of third-order and first-order agents. The interaction topology includes both fixed and switching cases. First, by a model transformation, heterogeneous multi-agent systems are converted into equivalent error systems. Then we analyze the consensus problem of the multi-agent systems by analyzing the stability problem of the error systems. For a fixed topology, a sufficient condition for consensus of heterogeneous multi-agent systems is obtained based on algebraic graph theory and linear system theory. For a switching topology, a necessary and sufficient condition for mean-square consensus of multi-agent systems is obtained based on algebraic graph theory and Markovian jump system theory. Finally, we give some simulation examples.

scholarly journals Making Walks Count: From Silent Circles to Hamiltonian Cycles

2017 ◽  
Author(s):  
Max A. Alekseyev ◽  
Gérard P. Michon
Keyword(s):  
Graph Theory ◽  
Hamiltonian Cycles ◽  
Algebraic Graph Theory ◽  
Counting Problems ◽  
Line Segments ◽  
Land Mass ◽  
Leonhard Euler ◽  
The City

Leonhard Euler (1707–1783) invented graph theory in 1735, by solving a puzzle of interest to the inhabitants of Königsberg. The city comprised three distinct land masses, connected by seven bridges. The residents sought a walk through the city that crossed each bridge exactly once but were consistently unable to find one. Euler showed that such a puzzle would have a solution if and only if every land mass was at the origin of an even number of bridges, with at most two exceptions—which could only be at the start or the end of the journey. Modern treatments of the problem capture Euler's reasoning by employing a diagram in which the land masses are represented by dots (called nodes), while the bridges are represented by line segments connecting the nodes (called edges). Such a diagram is referred to as a graph. This chapter uses algebraic graph theory to solve a number of counting problems.

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