Spectral extremal graphs for the bowtie

TL;DR

This paper characterizes the spectral extremal graphs avoiding the bowtie (F_2) for all sufficiently large n, identifying the unique extremal structure and extending classical extremal results to spectral settings.

Contribution

It removes the large n condition for the bowtie case and determines the unique spectral extremal graphs for all n ≥ 7, also analyzing the case with a fixed number of edges.

Findings

Unique extremal graph for F_2-free graphs when n ≥ 7 is a balanced bipartite with an extra edge.

For fixed edges m ≥ 8, the extremal graph is a join of K_2 with an independent set.

The result tightens previous bounds and generalizes classical extremal theorems to spectral extremal problems.

Abstract

Let $F_k$ be the (friendship) graph obtained from $k$ triangles by sharing a common vertex. The $F_k$-free graphs of order $n$ which attain the maximal spectral radius was firstly characterized by Cioab\u{a}, Feng, Tait and Zhang [Electron. J. Combin. 27 (4) (2020)], and later uniquely determined by Zhai, Liu and Xue [Electron. J. Combin. 29 (3) (2022)] under the condition that $n$ is sufficiently large. In this paper, we get rid of the condition on $n$ being sufficiently large if $k=2$. The graph $F_2$ is also known as the bowtie. We show that the unique $n$-vertex $F_2$-free spectral extremal graph is the balanced complete bipartite graph adding an edge in the vertex part with smaller size if $n\ge 7$, and the condition $n\ge 7$ is tight. Our result is a spectral generalization of a theorem of Erd\H{o}s, F\"{u}redi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995)], which states that $\mathrm{ex}(n,F_2)=\left\lfloor {n^2}/{4} \right\rfloor +1$. Moreover, we study the spectral extremal problem for $F_k$-free graphs with given number of edges. In particular, we show that the unique $m$-edge $F_2$-free spectral extremal graph is the join of $K_2$ with an independent set of $\frac{m-1}{2}$ vertices if $m\ge 8$, and the condition $m\ge 8$ is tight.