### Abstract

The second smallest eigenvalue of the Laplacian matrix, also known as the algebraic connectivity, characterizes the performance of some dynamic processes on networks, such as consensus in multiagent networks, synchronization of coupled oscillators, random walks on graphs, and so on. In a multiagent network, for example, the larger the algebraic connectivity of the graph representing interactions between agents is, the faster the convergence speed of a representative consensus algorithm is. This paper tackles the problem of finding graphs that maximize or locally maximize the algebraic connectivity in the space of graphs with a fixed number of vertices and edges. It is shown that some well-known classes of graphs such as star graphs, cycle graphs, complete bipartite graphs and circulant graphs are algebraic connectivity maximizers or local maximizers under certain conditions.

Original language | English |
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Article number | 7336529 |

Journal | IEEE Transactions on Control of Network Systems |

Volume | PP |

Issue number | 99 |

DOIs | |

Publication status | Published - 2015 |

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### Keywords

- Algebraic connectivity
- Consensus algorithm
- Convergence rate
- Laplacian matrix
- Multiagent network

### ASJC Scopus subject areas

- Control and Systems Engineering
- Signal Processing
- Computer Networks and Communications
- Control and Optimization

### Cite this

*IEEE Transactions on Control of Network Systems*,

*PP*(99), [7336529]. https://doi.org/10.1109/TCNS.2015.2503561

**Maximizing algebraic connectivity in the space of graphs with fixed number of vertices and edges.** / Ogiwara, Kohnosuke; Fukami, Tatsuya; Takahashi, Norikazu.

Research output: Contribution to journal › Article

*IEEE Transactions on Control of Network Systems*, vol. PP, no. 99, 7336529. https://doi.org/10.1109/TCNS.2015.2503561

}

TY - JOUR

T1 - Maximizing algebraic connectivity in the space of graphs with fixed number of vertices and edges

AU - Ogiwara, Kohnosuke

AU - Fukami, Tatsuya

AU - Takahashi, Norikazu

PY - 2015

Y1 - 2015

N2 - The second smallest eigenvalue of the Laplacian matrix, also known as the algebraic connectivity, characterizes the performance of some dynamic processes on networks, such as consensus in multiagent networks, synchronization of coupled oscillators, random walks on graphs, and so on. In a multiagent network, for example, the larger the algebraic connectivity of the graph representing interactions between agents is, the faster the convergence speed of a representative consensus algorithm is. This paper tackles the problem of finding graphs that maximize or locally maximize the algebraic connectivity in the space of graphs with a fixed number of vertices and edges. It is shown that some well-known classes of graphs such as star graphs, cycle graphs, complete bipartite graphs and circulant graphs are algebraic connectivity maximizers or local maximizers under certain conditions.

AB - The second smallest eigenvalue of the Laplacian matrix, also known as the algebraic connectivity, characterizes the performance of some dynamic processes on networks, such as consensus in multiagent networks, synchronization of coupled oscillators, random walks on graphs, and so on. In a multiagent network, for example, the larger the algebraic connectivity of the graph representing interactions between agents is, the faster the convergence speed of a representative consensus algorithm is. This paper tackles the problem of finding graphs that maximize or locally maximize the algebraic connectivity in the space of graphs with a fixed number of vertices and edges. It is shown that some well-known classes of graphs such as star graphs, cycle graphs, complete bipartite graphs and circulant graphs are algebraic connectivity maximizers or local maximizers under certain conditions.

KW - Algebraic connectivity

KW - Consensus algorithm

KW - Convergence rate

KW - Laplacian matrix

KW - Multiagent network

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U2 - 10.1109/TCNS.2015.2503561

DO - 10.1109/TCNS.2015.2503561

M3 - Article

AN - SCOPUS:85027516330

VL - PP

JO - IEEE Transactions on Control of Network Systems

JF - IEEE Transactions on Control of Network Systems

SN - 2325-5870

IS - 99

M1 - 7336529

ER -