Degree sequences of spectrally extremal connected nonregular graphs with prescribed maximum degree: the dense case
Year: 2026
Type: preprint
Abstract: Let $G$ be an $n$-vertex connected nonregular graph with maximum degree $Δ$ that attains the maximum spectral radius. For $3\leΔ\le n-2$, Liu and Li (2008) conjectured that $G$ has the degree sequence $(Δ,\ldots,Δ,δ)$ with $δ<Δ$ as large as possible. In this paper, we study the dense case where $Δ=n-k$ with $k\ge4$ fixed and $n$ sufficiently large. We prove that \[ d(G)= \begin{cases} (Δ,\ldots,Δ, 1), & \text{if $n$ is even and $k$ is odd};\\ (Δ,\ldots,Δ,Δ-1), & \text{if $n$ is odd and $k$ is even};\\ (Δ,\ldots,Δ,2), & \text{if $n$ and $k$ have the same parity}. \end{cases} \]
Source: arXiv (Cornell University)
Authors: Huang, Zejun, Yang, Chenxi
Institution: Shenzhen University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220458490