The spectral radius of graphs without trees of diameter at most four

TL;DR

This paper proves a spectral graph theory conjecture for large graphs, showing that graphs with high spectral radius contain all small-diameter trees unless they are a specific join graph.

Contribution

It confirms Nikiforov's conjecture for trees of diameter at most four, advancing understanding of spectral conditions for tree containment.

Findings

Conjecture holds for trees with diameter ≤ 4

High spectral radius implies presence of all such trees

Identifies the extremal graph as a join of complete and empty graphs

Abstract

Nikiforov (LAA, 2010) conjectured that for given integer $k$, any graph $G$ of sufficiently large order $n$ with spectral radius $\mu(G)\geq \mu(S_{n,k})$ contains all trees of order $2k+2$, unless $G=S_{n,k}$, where $S_{n,k}=K_k\vee \overline{K_{n-k}}$, the join of a complete graph of order $k$ and an empty graph of order $n-k$. In this paper, we show that the conjecture is true for trees of diameter at most four.