arXiv subjects

Yinfen Zhu

Publications and source records attributed to Yinfen Zhu.

3 recordsLinked to original sources

arXiv2026

Spectral extremal results for triangle-free graphs with chromatic number at least four

A graph is called $F$-free if it does not contain a copy of $F$. Let $G(r,s)$ denote a $K_{r+1}$-free graph of order $n$ with chromatic number at least $s$ that maximizes the spectral radius. Nikiforov [Linear Algebra Appl., 2007] proved the spectral Turán theorem, which implies that $G(r,s)$ is the $r$-partite Turán graph $T_{n,r}$ for $s\leq r$. Lin, Ning, and Wu [Combin. Probab. Comput., 2021] characterized the unique spectral extremal graph $G(2,3)$. This result was later extended by Li and Peng [SIAM J. Discrete Math., 2023] to all $s=r+1\geq 3$. In this paper, we push the characterization further by determining the unique extremal graph $G(2,4)$ for all sufficiently large $n$. Specifically, we show that $G(2,4)$ is precisely a blow-up of the Grötzsch graph. Interestingly, under the same conditions, $G(2,4)$ also coincides with the unique edge-extremal graph identified by Ren, Wang, Wang, and Yang [arXiv:2404.07486v2].

math.CO↗

arXiv2025

A note on the spectral radius and $[a,b]$-factor of graphs

The investigation of eigenvalue conditions for the existence of an $[a,b]$-factor originates in the work of Brouwer and Haemers (2005) on perfect matchings. In the decades since, spectral extremal problems related to $[a,b]$-factors have attracted considerable attention. In this paper, we establish a spectral radius condition that ensures the existence of an $[a,b]$-factor in a graph $G$ with minimum degree $δ(G) \geq a$, where $b > a \geq 1$. This result resolves a problem posed by Hao and Li [Electron. J. Combin. (2024)].

math.SP↗

arXiv2023

The least distance eigenvalue of the complements of graphs of diameter greater than three

Suppose $G$ is a connected simple graph with the vertex set $V( G ) = \{ v_1,v_2,\cdots ,v_n \} $. Let $d_G( v_i,v_j ) $ be the least distance between $v_i$ and $v_j$ in $G$. Then the distance matrix of $G$ is $D( G ) =( d_{ij} ) _{n\times n}$, where $d_{ij}=d_G( v_i,v_j ) $. Since $D( G )$ is a non-negative real symmetric matrix, its eigenvalues can be arranged as $λ_1(G)\ge λ_2(G)\ge \cdots \ge λ_n(G)$, where eigenvalue $λ_n(G)$ is called the least distance eigenvalue of $G$. In this paper we determine the unique graph whose least distance eigenvalue attains maximum among all complements of graphs of diameter greater than three.

math.CO↗