On a conjecture of spectral extremal problems

TL;DR

This paper proves a conjecture linking extremal graphs with maximum edges and spectral radius, showing that for large graphs avoiding a certain subgraph, the spectral extremal graph coincides with the edge extremal graph.

Contribution

It establishes that for graphs avoiding a specific subgraph, the spectral extremal graph is among the edge extremal graphs, confirming a conjecture for a broad class of graphs.

Findings

Spectral extremal graphs are contained within edge extremal graphs for large n.

The conjecture by Cioab, Desai, and Tait is fully proven.

The result applies to graphs obtained from Turán graphs by adding a bounded number of edges.

Abstract

For a simple graph $F$, let $\mathrm{Ex}(n, F)$ and $\mathrm{Ex_{sp}}(n,F)$ denote the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an $n$-vertex graph without any copy of the graph $F$, respectively. The Tur\'an graph $T_{n,r}$ is the complete $r$-partite graph on $n$ vertices where its part sizes are as equal as possible. Cioab\u{a}, Desai and Tait [The spectral radius of graphs with no odd wheels, European J. Combin., 99 (2022) 103420] posed the following conjecture: Let $F$ be any graph such that the graphs in $\mathrm{Ex}(n,F)$ are Tur\'{a}n graphs plus $O(1)$ edges. Then $\mathrm{Ex_{sp}}(n,F)\subset \mathrm{Ex}(n,F)$ for sufficiently large $n$. In this paper we consider the graph $F$ such that the graphs in $\mathrm{Ex}(n, F)$ are obtained from $T_{n,r}$ by adding $O(1)$ edges, and prove that if $G$ has the maximum spectral radius among all $n$-vertex graphs not containing $F$, then $G$ is a member of $\mathrm{Ex}(n, F)$ for $n$ large enough. Then Cioab\u{a}, Desai and Tait's conjecture is completely solved.