A Complete Solution to the Cvetković-Rowlinson Conjecture

Abstract

In 1990, Cvetković and Rowlinson [The largest eigenvalue of a graph: a survey, Linear Multilinear Algebra 28(1-2) (1990), 3--33] conjectured that among all outerplanar graphs on $n$ vertices, $K_1\vee P_{n-1}$ attains the maximum spectral radius. In 2017, Tait and Tobin [Three conjectures in extremal spectral graph theory, J. Combin. Theory, Ser. B 126 (2017) 137-161] confirmed the conjecture for sufficiently large values of $n$. In this article, we show the conjecture is true for all $n\ge 2$ except for $n=6$.

Since the conjecture is solved completely now, we change the title into "A Complete Solution to the Cvetković-Rowlinson Conjecture". 9 pages