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A Cheeger inequality for the lower spectral gap
Authors
Jyoti Prakash Saha
Publication date
22 June 2023
Publisher
View
on
arXiv
Abstract
Let
Γ
\Gamma
Γ
be a Cayley graph, or a Cayley sum graph, or a twisted Cayley graph, or a twisted Cayley sum graph, or a vertex-transitive graph. Denote the degree of
Γ
\Gamma
Γ
by
d
d
d
, its edge Cheeger constant by
h
Γ
\mathfrak{h}_\Gamma
h
Γ
​
, and its vertex Cheeger constant by
h
Γ
h_\Gamma
h
Γ
​
. Assume that
Γ
\Gamma
Γ
is undirected, non-bipartite. We prove that the edge bipartiteness constant of
Γ
\Gamma
Γ
is
Ω
(
h
Γ
/
d
)
\Omega({\mathfrak{h}_\Gamma}/{d})
Ω
(
h
Γ
​
/
d
)
, the vertex bipartiteness constant of
Γ
\Gamma
Γ
is
Ω
(
h
Γ
)
\Omega(h_\Gamma)
Ω
(
h
Γ
​
)
, and the smallest eigenvalue of the normalized adjacency operator of
Γ
\Gamma
Γ
is
−
1
+
Ω
(
h
Γ
2
/
d
2
)
-1 + \Omega({h_\Gamma^2}/{d^2})
−
1
+
Ω
(
h
Γ
2
​
/
d
2
)
. This answers in the affirmative a question of Moorman, Ralli and Tetali on the lower spectral gap of Cayley sum graphs
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oai:arXiv.org:2306.04436
Last time updated on 26/06/2023