Tight Estimates for Eigenvalues of Regular Graphs

Tight Estimates for Eigenvalues of Regular Graphs

Tight estimates for eigenvalues of regular graphs

It is shown that if a d-regular graph contains s vertices so that the distance between any pair is at least 4k, then its adjacency matrix has at least s eigenvalues which are at least 2 √ d − 1 cos( π 2k ). A similar result has been proved by Friedman using more sophisticated tools.

Regular Factors of Regular Graphs from Eigenvalues

Let r and m be two integers such that r > m. Let H be a graph with order |H|, size e and maximum degree r such that 2e > |H|r−m. We find a best lower bound on spectral radius of graph H in terms of m and r. Let G be a connected r-regular graph of order |G| and k < r be an integer. Using the previous results, we find some best upper bounds (in terms of r and k) on the third largest eigenvalue th...

Tight Distance-Regular Graphs

We consider a distance regular graph with diameter d and eigenvalues k d We show the intersection numbers a b satisfy k a d k a ka b a We say is tight whenever is not bipartite and equality holds above We charac terize the tight property in a number of ways For example we show is tight if and only if the intersection numbers are given by certain rational expressions involving d independent para...

On the extreme eigenvalues of regular graphs

In this paper, we present an elementary proof of a theorem of Serre concerning the greatest eigenvalues of k-regular graphs. We also prove an analogue of Serre’s theorem regarding the least eigenvalues of k-regular graphs: given > 0, there exist a positive constant c = c( , k) and a nonnegative integer g = g( , k) such that for any k-regular graph X with no odd cycles of length less than g, the...