The maximum spectral radius of planner graphs without the joint of K2 and a linear forest

Given a graph F, let SPEXP(n,F) be the set of graphs with the maximum spectral radius among all F-free n-vertex planner graph. In 2017, Tait and Tobin proved that for sufficiently n, K₂+Pₙ₋₂ is the unique graph with the maximum spectral radius over all n-vertex planner graphs. In this paper, focusing on SPEXP(n,K₂+H) in which H is a linear forest, we prove that SPEXP(n,K₂+H)=\2K₁+Cₙ₋₂\ when H∈ ₂,P₃,Iq\ (p≥1, q≥ 3), where Kₙ, Pₙ, Iₙ are complete graph, path and empty graph of order n, respectively. When H contains a P₄, we prove that 2K₁+Cₙ₋₂∉ SPEXP(n,K₂+H) and also provide a structural characterization of graphs in SPEXP(n,K₂+H).