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 + Pₙ₋₂ is the unique graph attaining the maximum spectral radius among all planar graphs on n vertices, where K₂ + Pₙ₋₂ is the graph obtained from K₂∪ Pₙ₋₂ by adding all possible edges between K₂ and Pₙ₋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 SPEXP(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 SPEXP(n,F) when F is a planar graph not contained in K2,n-2. As applications, we determine the extremal graphs in SPEXP(n,Wₖ), SPEXP(n,Fₖ) and SPEXP(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.