Characterization of Trivalent Graphs with Minimal Eigenvalue Gap*
نویسندگان
چکیده
Among all trivalent graphs on n vertices, let Gn be one with the smallest possible eigenvalue gap. (The eigenvalue gap is the difference between the two largest eigenvalues of the adjacency matrix; for regular graphs, it equals the second smallest eigenvalue of the Laplacian matrix.) We show that Gn is unique for each n and has maximum diameter. This extends work of Guiduli and solves a conjecture implicit in a paper of Bussemaker, ^obelji}, Cvetkovi} and Seidel. Depending on n, the graph Gn may not be the only one with maximum diameter. We thus also determine all cubic graphs with maximum diameter for a given number n of vertices.
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