Emerging Applications of Number Theory pp 387-403 | Cite as

# Eigenvalues of Ramanujan Graphs

Chapter

## Abstract

The adjacency matrix of a (directed or undirected) finite graph where Assume

*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$$(Af)(x) = \sum\limits_{y} {f(y),}$$

*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$$\lambda = \lambda (X){{ = }_{{\lambda i}}}\begin{array}{*{20}{c}} {\max } \\ {nontrivial} \\ \end{array} |{{\lambda }_{i}}|.$$

*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.## Keywords

Cayley Graph Automorphic Form Congruence Subgroup Double Coset Quaternion Algebra
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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