By Türker Biyikoğu, Josef Leydold, Peter F. Stadler (auth.)

Eigenvectors of graph Laplacians haven't, thus far, been the topic of expository articles and therefore they might appear a shocking subject for a e-book. The authors suggest motivations for this new LNM quantity: (1) There are attention-grabbing sophisticated ameliorations among the houses of suggestions of Schrödinger equations on manifolds at the one hand, and their discrete analogs on graphs. (2) "Geometric" homes of (cost) features outlined at the vertex units of graphs are of sensible curiosity for heuristic optimization algorithms. The remark that the associated fee services of a variety of of the well-studied combinatorial optimization difficulties are eigenvectors of linked graph Laplacians has triggered the research of such eigenvectors.

The quantity investigates the constitution of eigenvectors and appears on the variety of their signal graphs ("nodal domains"), Perron parts, graphs with extremal houses with recognize to eigenvectors. The Rayleigh quotient and rearrangement of graphs shape the most methodology.

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**Extra resources for Laplacian Eigenvectors of Graphs: Perron-Frobenius and Faber-Krahn Type Theorems**

**Sample text**

However, for convenience Fiedler [65] has set v(Kn ) = n − 1. 10. The bound in Thm. 8 can be improved. Kirkland gives a sharp upper bound for the algebraic connectivity in terms of the number n of vertices and the number of cutpoints of the graph, see [112, 116]. There is also a lower bound for the algebraic connectivity. The proof uses a result for symmetric doubly stochastic matrices, see [64]. 11 ([65]). For any graph G with n vertices λ2 (G) ≥ 2 1 − cos Equality holds if and only if G is a path.

There is also a lower bound for the algebraic connectivity. The proof uses a result for symmetric doubly stochastic matrices, see [64]. 11 ([65]). For any graph G with n vertices λ2 (G) ≥ 2 1 − cos Equality holds if and only if G is a path. π e(G) . 11 can be generalized for Laplacians on weighted graphs, see [68]. An eigenfunction aﬀording the second smallest eigenvalue of a graph Laplacian is called a Fiedler vector of G. As mentioned in the introduction, Fiedler vectors have many applications in various ﬁelds of application, where they form the basis of heuristic approaches to solve hard combinatorial optimization problems including graph bi-partitioning [146] and spectral clustering [15].

5 The Courant-Herrmann Conjecture A footnote on p. 454 in [44] claims that the nodal domain theorem for manifolds (p. 29) can be generalized in the following way: Any linear combination of the ﬁrst n eigenfunctions divides the domain, by means of its nodes, into no more than n subdomains. For a proof the reader is referred to Herrmann’s 1932 dissertation in G¨ ottingen. We will refer to this statement as the CourantHerrmann conjecture (CHC). Gladwell and Zhu [82] reported, however, that neither in this work nor in any of Herrmann’s subsequent publications such a result is stated, let alone proved.