The spectral even cycle problem

TL;DR

This paper proves a conjecture about the maximum spectral radius of large graphs avoiding certain even cycles, identifying specific extremal graphs for these conditions.

Contribution

It resolves a two-part conjecture by Nikiforov on the extremal spectral radii of graphs avoiding even cycles of specified lengths.

Findings

Confirmed the extremal graphs for $C_{2k+2}$-free graphs are $S_{n,k}^+$.

Identified the extremal graphs for graphs avoiding both $C_{2k+1}$ and $C_{2k+2}$ as $S_{n,k}$.

Provided proofs for large $n$ cases of the conjecture.

Abstract

In this paper, we study the maximum adjacency spectral radii of graphs of large order that do not contain an even cycle of given length. For $n>k$, let $S_{n,k}$ be the join of a clique on $k$ vertices with an independent set of $n-k$ vertices and denote by $S_{n,k}^+$ the graph obtained from $S_{n,k}$ by adding one edge. In 2010, Nikiforov conjectured that for $n$ large enough, the $C_{2k+2}$-free graph of maximum spectral radius is $S_{n,k}^+$ and that the $\{C_{2k+1},C_{2k+2}\}$-free graph of maximum spectral radius is $S_{n,k}$. We solve this two-part conjecture.