Abstract
The algebraic connectivity of a graph, introduced by Fiedler, is defined to be its second smallest Laplacian eigenvalue. It provides a measure of the degree of connectivity of the graph. We first prove basic properties of algebraic connectivity and the associated eigenvector, known as Fiedler vector. A classification of trees into two types, based on the Fiedler vector, is described. Some monotonocity properties of the coordinators of the Fiedler vector of a tree are proved. Bounds for the algebraic connectivity are obtained. These bounds are important in the study of random walks on the graph.
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References and Further Reading
Akbari, S., Kirkland, S.J.: On unimodular graphs. Linear Algebra Appl. 421, 3–15 (2007)
Bapat, R.B.: A note on singular line graphs. Bulletin Kerala Math. Assoc. 8(2), 207–209 (2011)
Bapat, R.B., Souvik Roy: On the adjacency matrix of a block graph, Linear and Multilinear Algebra, to appear
Ebrahim Ghorbani: Spanning trees and line graph eigenvalues, arXiv:1201.3221v1 (2012)
Ebrahim Ghorbani: Spanning trees and line graph eigenvalues, arXiv:1201.3221v3 (2013)
Ivan Gutman, Irene Sciriha: On the nullity of line graphs of trees. Discrete Math. 232, 35–45 (2001)
Irene Sciriha: On singular line graphs of trees. Congressus Numeratium 135, 73–91 (1998)
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Bapat, R.B. (2014). Line Graph of a Tree. In: Graphs and Matrices. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-6569-9_7
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DOI: https://doi.org/10.1007/978-1-4471-6569-9_7
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