A complete solution to the Boots-Royle/Cao-Vince conjecture

Abstract

Boots and Royle, and independently Cao and Vince, conjectured that the join of an edge and a path on $n-2$ vertices is the unique planar graph of maximum adjacency spectral radius for $n\geq 9$. Tait and Tobin (JCTB, 2017) proved the conjecture for sufficiently large order. In this paper, we completely resolved the Boots-Royle/Cao-Vince conjecture.

Disclosure

“= 3n − 6 − (2n − 3) = |V (G′ )| − 1. An acyclic graph on |V (G′ )| vertices with |V (G′ )| − 1 edges is connected. Therefore G′ = Pn−2 , and G = K2 ∨ Pn−2 . This proves the Theorem 1.1. Declaration on the use of AI The authors used ChatGPT to generate code for searching the extremal planar graphs for n ≤ 14, and to assist with several computations and symbolic derivations. ChatGPT was also used for grammar checking, language polishing, and improving the clarity of the exposi”

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Rewriting existing author-written text

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4

Verified

2026-08-23T19:00:58.280386Z

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