On the spectral extremal problem of planar graphs

The spectral extremal problem of planar graphs has aroused a lot of interest over the past three decades. In 1991, Boots and Royle Geogr. Anal. 23 (3) (1991) 276--282 (and Cao and Vince Linear Algebra Appl. 187 (1993) 251--257 independently) conjectured that K₂ + P₍-₂ is the unique graph attaining the maximum spectral radius among all planar graphs on n vertices, where K₂ + P₍-₂ is the graph obtained from K₂ P₍-₂ by adding all possible edges between K₂ and P₍-₂. In 2017, Tait and Tobin J. Combin. Theory Ser. B 126 (2017) 137--161 confirmed this conjecture for all sufficiently large n. In this paper, we consider the spectral extremal problem for planar graphs without specified subgraphs. For a fixed graph F, let SPEX₏ (n, F) denote the set of graphs attaining the maximum spectral radius among all F-free planar graphs on n vertices. We describe a rough sturcture for the connected extremal graphs in SPEX₏ (n, F) when F is a planar graph not contained in K₂, ₍-₂. As applications, we determine the extremal graphs in SPEX₏ (n, Wₖ), SPEX₏ (n, Fₖ) and SPEX₏ (n, (k+1) K₂) for all sufficiently large n, where Wₖ, Fₖ and (k+1) K₂ are the wheel graph of order k, the friendship graph of order 2k+1 and the disjoint union of k+1 copies of K₂, respectively.