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Variants of spectral Turán theorems and eigenvectors of graphs · arXivDesk

2312.16138Dec 26, 202320 pages. This is a new version of the previous paper titled "A local version of spectral Turán theorem". In this version, we add more results, including a variant of spectral Turán theorem and a disproof of a conjecture of Gregory

Variants of spectral Turán theorems and eigenvectors of graphs

Lele Liu, Bo Ning

Combinatorics

arXiv PDF

Abstract

In 2002, Nikiforov proved that for an $n$ -vertex graph $G$ with clique number $ω$ and edge number $m$ , the spectral radius $λ(G)$ satisfies $λ(G) \leq \sqrt{2(1 - 1/ω) m}$

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, which confirmed a conjecture implicitly suggested by Edwards and Elphick. In this paper, we prove a local version of spectral Turán inequality, which states that

λ^2(G)\leq 2\sum_{e\in E(G)}\frac{c(e)-1}{c(e)}

, where

c(e)

is the order of the largest clique containing the edge

e

G

. We also characterize the extremal graphs. We prove that our theorem implies Nikiforov's theorem and give an example to show that the difference of Nikiforov's bound and ours is

Ω(\sqrt{m})

for some cases. Additionally, we establish a spectral counterpart to Ore's problem (1962) which asks for the maximum size of an

n

-vertex graph such that its complement is connected and does not contain

F

as a subgraph. Our result leads to a new spectral Turán inequality applicable to graphs with connected complements. Finally, we disprove a conjecture of Gregory, asserting that for a connected

n

-vertex graph

G

with chromatic number

k\geq 2

and an independent set

S

, we have

\sum_{v\in S} x_v^2 \leq \frac{1}{2} - \frac{k-2}{2\sqrt{(k-2)^2 + 4(k-1)(n-k+1)}},

where

x_v

is the component of the Perron vector of

G

with respect to the vertex

v

. A modified version of Gregory's conjecture is proposed.

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