# algebraic connectivity of a graph

Let $L(G)$ be the Laplacian matrix (http://planetmath.org/LaplacianMatrixOfAGraph) of a finite connected graph $G$ with $n$ vertices. Let the eigenvalues of $L(G)$ be denoted by $\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}$, which is the usual notation in spectral graph theory. The connectivity of $G$ is $\lambda_{2}$. The usual notation for the algebraic connectivity is $a(G)$. The parameter is a measure of how well the graph is connected. For example, $a(G)\not=0$ if and only if $G$ is connected.

## References

• 1 Fieldler, M. Algebraic connectivity of graphs, Czech. Math. J. 23 (98) (1973) pp. 298-305.
• 2 Merris, R. Laplacian matrices of graphs: a survey, Lin. Algebra and its Appl. 197/198 (1994) pp. 143-176.
Title algebraic connectivity of a graph AlgebraicConnectivityOfAGraph 2013-03-22 17:04:37 2013-03-22 17:04:37 Mathprof (13753) Mathprof (13753) 9 Mathprof (13753) Definition msc 05C50 algebraic connectivity