Eigenvalues of Ramanujan Graphs

  • Wen-Ching Winnie Li
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 109)

Abstract

The adjacency matrix of a (directed or undirected) finite graph X is a square matrix A = A X with rows and columns parametrized by the vertices of X such that the xy entry of A records the number of edges from the vertex x to the vertex y. It may be regarded as a linear operator on the space of functions on vertices of X which sends a function f to Af whose value at a vertex x is given by
$$(Af)(x) = \sum\limits_{y} {f(y),}$$
where y runs through all ending points of the edges starting from x. The eigenvalues of A are called the eigenvalues of X. Call X a k-regular graph if at every vertex of X there are k edges going out and k edges coming in. If X is undirected and k-regular, then the eigenvalues of X are real and bounded between k and -k. The multiplicity of k as an eigenvalue of X is equal to the number of connected components of X, and -k is an eigenvalue if and only if X is a bipartite graph. In general, the eigenvalues λ i of X provide lots of information about the graph G. An eigenvalue of X with absolute value k is called a trivial eigenvalue, we are interested in the nontrivial eigenvalues. For this matter, define
$$\lambda = \lambda (X){{ = }_{{\lambda i}}}\begin{array}{*{20}{c}} {\max } \\ {nontrivial} \\ \end{array} |{{\lambda }_{i}}|.$$
Assume X connected. It turns out that the smaller λ(X) is, the larger the magnifying constant X has, and hence the more ”expanding” X is. It is also known that the smaller λ(X) is, the smaller the diameter of X is. In the case that X represents a communication network, the transmission efficiency is measured by the magnifying constant, and the transmission delay is measured by the diameter. Thus regular graphs with small λ have important applications in communication networks and in computer science.

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Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Wen-Ching Winnie Li
    • 1
  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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