Algebraic Connectivity
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The algebraic connectivity (also known as Fiedler value or Fiedler eigenvalue after
Miroslav Fiedler Miroslav Fiedler (7 April 1926 – 20 November 2015) was a Czech mathematician known for his contributions to linear algebra, graph theory and algebraic graph theory. His article, "Algebraic Connectivity of Graphs", published in the ''Czechoslova ...
) of a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
''G'' is the second-smallest
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
(counting multiple eigenvalues separately) of the
Laplacian matrix In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Lapl ...
of ''G''. This eigenvalue is greater than 0 if and only if ''G'' is a
connected graph In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgra ...
. This is a corollary to the fact that the number of times 0 appears as an eigenvalue in the Laplacian is the number of connected components in the graph. The magnitude of this value reflects how well connected the overall graph is. It has been used in analyzing the robustness and
synchronizability Synchronization is the coordination of events to operate a system in unison. For example, the conductor of an orchestra keeps the orchestra synchronized or ''in time''. Systems that operate with all parts in synchrony are said to be synchronou ...
of networks.


Properties

The truncated icosahedron or connectivity of 3, and an algebraic connectivity of 0.243.">Buckminsterfullerene">truncated icosahedron or Buckminsterfullerene graph has a traditional connectivity (graph theory)">connectivity of 3, and an algebraic connectivity of 0.243. The algebraic connectivity of undirected graphs with nonnegative weights, a(G)\geq0 with the inequality being strict if and only if G is connected. However, the algebraic connectivity can be negative for general directed graphs, even if ''G'' is a
connected graph In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgra ...
. Furthermore, the value of the algebraic connectivity is bounded above by the traditional (vertex)
connectivity of the graph, \text \le \text. If the number of vertices of an undirected connected graph with nonnegative edge weights is ''n'' and the diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
is ''D'', the algebraic connectivity is also known to be bounded below by \text \ge \frac , and in fact (in a result due to Brendan McKay) by \frac. Brendan McKay (mathematician)">Brendan McKay) by \frac.Bojan Mohar
The Laplacian Spectrum of Graphs
in ''Graph Theory, Combinatorics, and Applications'', Vol. 2, Ed. Y. Alavi, G. Chartrand, Ortrud Oellermann">O. R. Oellermann, A. J. Schwenk, Wiley, 1991, pp. 871–898.
For the graph with 6 nodes show above (n=6,D=3) these bound means, 4/18 = 0.222 ≤ algebraic connectivity 0.722  ≤ connectivity 1. Unlike the traditional connectivity, the algebraic connectivity is dependent on the number of vertices, as well as the way in which vertices are connected. In random graphs, the algebraic connectivity decreases with the number of vertices, and increases with the average degree (graph theory), degree. The exact definition of the algebraic connectivity depends on the type of Laplacian used. Fan Chung has developed an extensive theory using a rescaled version of the Laplacian, eliminating the dependence on the number of vertices, so that the bounds are somewhat different. In models of synchronization on networks, such as the Kuramoto model, the Laplacian matrix arises naturally, so the algebraic connectivity gives an indication of how easily the network will synchronize. Other measures, such as the average
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
(characteristic path length) can also be used, and in fact the algebraic connectivity is closely related to the (reciprocal of the) average distance. The algebraic connectivity also relates to other connectivity attributes, such as the isoperimetric number, which is bounded below by half the algebraic connectivity.


Fiedler vector

The original theory related to algebraic connectivity was produced by
Miroslav Fiedler Miroslav Fiedler (7 April 1926 – 20 November 2015) was a Czech mathematician known for his contributions to linear algebra, graph theory and algebraic graph theory. His article, "Algebraic Connectivity of Graphs", published in the ''Czechoslova ...
.M. Fiedler. "Laplacian of graphs and algebraic connectivity", Combinatorics and Graph Theory (Warsaw, 1987), ''Banach Center Publications'' 25(1) (1989), 57–70. In his honor the
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
associated with the algebraic connectivity has been named the Fiedler vector. The Fiedler vector can be used to partition a graph.


Partitioning a graph using the Fiedler vector

For the example graph in the introductory section, the Fiedler vector is \begin 0.415 & 0.309 & 0.069 & -0.221 & 0.221 & -0.794 \end . The negative values are associated with the poorly connected vertex 6, and the neighbouring articulation point, vertex 4; while the positive values are associated with the other vertices. The ''signs of the values in the Fiedler vector'' can therefore be used to partition this graph into two components: \, \ . Alternatively, the value of 0.069 (which is close to zero) can be placed in a class of its own, partitioning the graph into three components: \, \ , \ or moved to the other partition \, \ , as pictured. The squared values of the components of the Fiedler vector, summing up to one since the vector is normalized, can be interpreted as probabilities of the corresponding data points to be assigned to the sign-based partition.


See also

*
Connectivity (graph theory) In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subg ...
* Graph property


References

{{reflist Algebraic graph theory Graph connectivity Graph invariants