The structure of trivalent graphs with minimal eigenvalue gap (Q1370464)

Let \(G\) be a connected cubic graph on \(n\) vertices \((n\geq 10)\) with a maximal second largest eigenvalue (of the adjacency matrix). It is shown that \(G\) is reduced path-like what means that the block intersection graph of \(G\) is a path with blocks having 2, 4, 5, 6 or 7 vertices. This result partially verifies an implicit conjecture of \textit{F. C. Bussemaker}, \textit{S. Čobeljić}, the reviewer and \textit{J. J. Seidel} [J. Comb. Theory, Ser. B 23, 234-235 (1977; Zbl 0369.05045)].