A Complete Solution to the Cvetkovi\'{c}-Rowlinson Conjecture

TL;DR

This paper proves the Cvetković-Rowlinson conjecture that a specific outerplanar graph maximizes spectral radius among all such graphs, confirming it for all but one small case.

Contribution

The paper completes the proof of the conjecture for all n ≥ 2, except for n=6, resolving a long-standing open problem in spectral graph theory.

Findings

Confirmed the conjecture for all n ≥ 2 except n=6.

Identified the extremal graph as K_1 ∨ P_{n-1}.

Resolved the conjecture for the entire class of outerplanar graphs.

Abstract

In 1990, Cvetkovi\'{c} 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 $n\geq2$ except for $n=6$.