Abstract
We consider regular graphs with small second largest eigenvalue (denoted by ?2). In particular, we determine all triangle-free regular graphs with ?2 ? ?2, all bipartite regular graphs with ?2 ? ?3, and all bipartite regular graphs of degree 3 with ?2 ? 2.
Highlights
The characteristic polynomial and the eigenvalues of a simple graph G are defined as the characteristic polynomial and the eigenvalues of its adjacency matrix
Regular graphs with small second largest eigenvalue often have high connectivity properties, and they are relevant to theoretical computer science, the designs of robust computer networks, the theory of error correcting codes, and to complexity theory [8]
In our previous work we considered regular graphs with λ2 ≤ 1
Summary
Regular graphs with small second largest eigenvalue often have high connectivity properties, and they are relevant to theoretical computer science, the designs of robust computer networks, the theory of error correcting codes, and to complexity theory [8]. We increase this√bound, and we completely determine all t√riangle-free regular graphs with λ2 ≤ 2, all bipartite regular graphs with λ2 ≤ 3, and all bipartite reflexive cubic (i.e. regular graphs of degree 3) graphs. We use both theoretical and computational methods; for the computer search we use the GENREG
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