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Abstract Given a planar graph family , let and be the maximum size and maximum spectral radius over all ‐vertex ‐free planar graphs, respectively. Let be the disjoint union of copies of ‐cycles, and be the family of vertex‐disjoint cycles without length restriction. Tait and Tobin determined that is the extremal spectral graph among all planar graphs with sufficiently large order , which implies the extremal graphs of both and for are . In this paper, we first determine and and characterize the unique extremal graph for , and sufficiently large . Second, we obtain the exact values of and , which solve a conjecture of Li for .
Fang et al. (2024) studied this question.