The Structure of Trivalent Graphs with Minimal Eigenvalue Gap

Let G be a connected trivalent graph on n vertices (n ≥ 10) such that among all connected trivalent graphs on n vertices, G has the largest possible second eigenvalue. We show that G must be reduced path-like, i.e. G must be of the form: where the ends are one of the following: q q q q q @ @ @ @ @ or q q @ @ @ q q q q @ @q (the right-hand end block is the mirror image of one of the blocks shown) and the middle blocks are one of the following: q @ @q q @ @q or q @ @q q q q @ @q This partially solves a conjecture implicit in a paper of Bussemaker, Čobeljić, Cvetković, and Seidel [3].