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The algebraic connectivity (also known as Fiedler value or Fiedler eigenvalue after Miroslav Fiedler) 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 denoted b ...
(counting multiple eigenvalues separately) of the Laplacian matrix 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 subgrap ...
. 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 of networks.


Properties

The truncated_icosahedron_or_Buckminsterfullerene_graph_has_a_traditional_connectivity_(graph_theory).html" "title="Buckminsterfullerene.html" ;"title="truncated icosahedron or Buckminsterfullerene">truncated icosahedron or Buckminsterfullerene graph has a traditional connectivity (graph theory)">connectivity Connectivity may refer to: Computing and technology * Connectivity (media), the ability of the social media to accumulate economic capital from the users connections and activities * Internet connectivity, the means by which individual terminals, ...
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 subgrap ...
. Furthermore, the value of the algebraic connectivity is bounded above by the traditional (vertex)
connectivity Connectivity may refer to: Computing and technology * Connectivity (media), the ability of the social media to accumulate economic capital from the users connections and activities * Internet connectivity, the means by which individual terminals, ...
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 for ...
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.
Bojan Mohar Bojan Mohar (born September 21, 1956) is a Slovenian and Canadian mathematician, working in graph theory. He is a professor of mathematics at the University of Ljubljana and the holder of a Canada Research Chair in graph theory at Simon Fraser Unive ...

The Laplacian Spectrum of Graphs
in ''Graph Theory, Combinatorics, and Applications'', Vol. 2, Ed. Y. Alavi, G. Chartrand, 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 graph In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs li ...
s, the algebraic connectivity decreases with the number of vertices, and increases with the average
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
. The exact definition of the algebraic connectivity depends on the type of Laplacian used.
Fan Chung Fan-Rong King Chung Graham (; born October 9, 1949), known professionally as Fan Chung, is a Taiwanese-born American mathematician who works mainly in the areas of spectral graph theory, extremal graph theory and random graphs, in particular in g ...
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 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 ...
on networks, such as the
Kuramoto model The Kuramoto model (or Kuramoto–Daido model), first proposed by , is a mathematical model used to describing synchronization. More specifically, it is a model for the behavior of a large set of coupled oscillators. Its formulation was motivated ...
, 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.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 denoted b ...
associated with the algebraic connectivity has been named the Fiedler vector. The Fiedler vector can be used to
partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
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 Articulation may refer to: Linguistics * Articulatory phonetics, the study of how humans produce speech sounds via the interaction of physiological structures ** Manner of articulation, how speech organs involved in making a sound make contact * ...
, 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 subgra ...
*
Graph property In graph theory, a graph property or graph invariant is a property of graphs that depends only on the abstract structure, not on graph representations such as particular labellings or drawings of the graph.. Definitions While graph drawing and ...


References

{{reflist Algebraic graph theory Graph connectivity Graph invariants