On the Second Eigenvalue of Random Bipartite Biregular Graphs
We consider the spectral gap of a uniformly chosen random ( d 1 , d 2 ) -biregular bipartite graph G with | V 1 | = n , | V 2 | = m , where d 1 , d 2 could possibly grow with n and m . Let A be the adjacency matrix of G . Under the assumption that d 1 ≥ d 2 and d 2 = O ( n 2 / 3 ) , we show that λ 2...
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| Veröffentlicht in: | Journal of theoretical probability 2023-06, Vol.36 (2), p.1269-1303 |
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| creator | Zhu, Yizhe |
| description | We consider the spectral gap of a uniformly chosen random
(
d
1
,
d
2
)
-biregular bipartite graph
G
with
|
V
1
|
=
n
,
|
V
2
|
=
m
, where
d
1
,
d
2
could possibly grow with
n
and
m
. Let
A
be the adjacency matrix of
G
. Under the assumption that
d
1
≥
d
2
and
d
2
=
O
(
n
2
/
3
)
,
we show that
λ
2
(
A
)
=
O
(
d
1
)
with high probability. As a corollary, combining the results from (Tikhomirov and Yousse in Ann Probab 47(1):362–419, 2019), we show that the second singular value of a uniform random
d
-regular digraph is
O
(
d
)
for
1
≤
d
≤
n
/
2
with high probability. Assuming
d
2
is fixed and
d
1
=
O
(
n
2
)
, we further prove that for a random
(
d
1
,
d
2
)
-biregular bipartite graph,
|
λ
i
2
(
A
)
-
d
1
|
=
O
(
d
1
)
for all
2
≤
i
≤
n
+
m
-
1
with high probability. The proofs of the two results are based on the size biased coupling method introduced in Cook et al. (Ann Probab 46(1):72–125, 2018) for random
d
-regular graphs and several new switching operations we define for random bipartite biregular graphs. |
| doi_str_mv | 10.1007/s10959-022-01190-0 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2817624141</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2817624141</sourcerecordid><originalsourceid>FETCH-LOGICAL-c319t-52772661a017402491ffe9c451ab1568c3b168eb4993491911e69dff83e2d0893</originalsourceid><addsrcrecordid>eNp9kM1LAzEQxYMoWKv_gKcFz9GZJPsxBw9aahUKBT_OId1N2i3t7prsCv73Rlfw5mkG5r03jx9jlwjXCJDfBARKiYMQHBAJOByxCaa54CQkHLMJFKQ4FQpO2VkIOwAgApiw21WT9FubvNiybapkXm9s82H2g01alzybpmoPyX3dGd_XvY2bt5thb3yy8KbbhnN24sw-2IvfOWVvD_PX2SNfrhZPs7slLyVSz1OR5yLL0ADmCoQidM5SqVI0a0yzopRrzAq7VkQyHgnRZlQ5V0grqthcTtnVmNv59n2wode7dvBNfKlFgXkmFCqMKjGqSt-G4K3Tna8Pxn9qBP2NSY-YdMSkfzBpiCY5mkIUNxvr_6L_cX0Be8Znyw</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2817624141</pqid></control><display><type>article</type><title>On the Second Eigenvalue of Random Bipartite Biregular Graphs</title><source>SpringerLink Journals</source><creator>Zhu, Yizhe</creator><creatorcontrib>Zhu, Yizhe</creatorcontrib><description>We consider the spectral gap of a uniformly chosen random
(
d
1
,
d
2
)
-biregular bipartite graph
G
with
|
V
1
|
=
n
,
|
V
2
|
=
m
, where
d
1
,
d
2
could possibly grow with
n
and
m
. Let
A
be the adjacency matrix of
G
. Under the assumption that
d
1
≥
d
2
and
d
2
=
O
(
n
2
/
3
)
,
we show that
λ
2
(
A
)
=
O
(
d
1
)
with high probability. As a corollary, combining the results from (Tikhomirov and Yousse in Ann Probab 47(1):362–419, 2019), we show that the second singular value of a uniform random
d
-regular digraph is
O
(
d
)
for
1
≤
d
≤
n
/
2
with high probability. Assuming
d
2
is fixed and
d
1
=
O
(
n
2
)
, we further prove that for a random
(
d
1
,
d
2
)
-biregular bipartite graph,
|
λ
i
2
(
A
)
-
d
1
|
=
O
(
d
1
)
for all
2
≤
i
≤
n
+
m
-
1
with high probability. The proofs of the two results are based on the size biased coupling method introduced in Cook et al. (Ann Probab 46(1):72–125, 2018) for random
d
-regular graphs and several new switching operations we define for random bipartite biregular graphs.</description><identifier>ISSN: 0894-9840</identifier><identifier>EISSN: 1572-9230</identifier><identifier>DOI: 10.1007/s10959-022-01190-0</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Eigenvalues ; Graph theory ; Graphs ; Mathematics ; Mathematics and Statistics ; Probability Theory and Stochastic Processes ; Statistics</subject><ispartof>Journal of theoretical probability, 2023-06, Vol.36 (2), p.1269-1303</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022</rights><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-52772661a017402491ffe9c451ab1568c3b168eb4993491911e69dff83e2d0893</citedby><cites>FETCH-LOGICAL-c319t-52772661a017402491ffe9c451ab1568c3b168eb4993491911e69dff83e2d0893</cites><orcidid>0000-0002-5665-0374</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10959-022-01190-0$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10959-022-01190-0$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Zhu, Yizhe</creatorcontrib><title>On the Second Eigenvalue of Random Bipartite Biregular Graphs</title><title>Journal of theoretical probability</title><addtitle>J Theor Probab</addtitle><description>We consider the spectral gap of a uniformly chosen random
(
d
1
,
d
2
)
-biregular bipartite graph
G
with
|
V
1
|
=
n
,
|
V
2
|
=
m
, where
d
1
,
d
2
could possibly grow with
n
and
m
. Let
A
be the adjacency matrix of
G
. Under the assumption that
d
1
≥
d
2
and
d
2
=
O
(
n
2
/
3
)
,
we show that
λ
2
(
A
)
=
O
(
d
1
)
with high probability. As a corollary, combining the results from (Tikhomirov and Yousse in Ann Probab 47(1):362–419, 2019), we show that the second singular value of a uniform random
d
-regular digraph is
O
(
d
)
for
1
≤
d
≤
n
/
2
with high probability. Assuming
d
2
is fixed and
d
1
=
O
(
n
2
)
, we further prove that for a random
(
d
1
,
d
2
)
-biregular bipartite graph,
|
λ
i
2
(
A
)
-
d
1
|
=
O
(
d
1
)
for all
2
≤
i
≤
n
+
m
-
1
with high probability. The proofs of the two results are based on the size biased coupling method introduced in Cook et al. (Ann Probab 46(1):72–125, 2018) for random
d
-regular graphs and several new switching operations we define for random bipartite biregular graphs.</description><subject>Eigenvalues</subject><subject>Graph theory</subject><subject>Graphs</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Probability Theory and Stochastic Processes</subject><subject>Statistics</subject><issn>0894-9840</issn><issn>1572-9230</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><recordid>eNp9kM1LAzEQxYMoWKv_gKcFz9GZJPsxBw9aahUKBT_OId1N2i3t7prsCv73Rlfw5mkG5r03jx9jlwjXCJDfBARKiYMQHBAJOByxCaa54CQkHLMJFKQ4FQpO2VkIOwAgApiw21WT9FubvNiybapkXm9s82H2g01alzybpmoPyX3dGd_XvY2bt5thb3yy8KbbhnN24sw-2IvfOWVvD_PX2SNfrhZPs7slLyVSz1OR5yLL0ADmCoQidM5SqVI0a0yzopRrzAq7VkQyHgnRZlQ5V0grqthcTtnVmNv59n2wode7dvBNfKlFgXkmFCqMKjGqSt-G4K3Tna8Pxn9qBP2NSY-YdMSkfzBpiCY5mkIUNxvr_6L_cX0Be8Znyw</recordid><startdate>20230601</startdate><enddate>20230601</enddate><creator>Zhu, Yizhe</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-5665-0374</orcidid></search><sort><creationdate>20230601</creationdate><title>On the Second Eigenvalue of Random Bipartite Biregular Graphs</title><author>Zhu, Yizhe</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-52772661a017402491ffe9c451ab1568c3b168eb4993491911e69dff83e2d0893</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Eigenvalues</topic><topic>Graph theory</topic><topic>Graphs</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Probability Theory and Stochastic Processes</topic><topic>Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Zhu, Yizhe</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of theoretical probability</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Zhu, Yizhe</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the Second Eigenvalue of Random Bipartite Biregular Graphs</atitle><jtitle>Journal of theoretical probability</jtitle><stitle>J Theor Probab</stitle><date>2023-06-01</date><risdate>2023</risdate><volume>36</volume><issue>2</issue><spage>1269</spage><epage>1303</epage><pages>1269-1303</pages><issn>0894-9840</issn><eissn>1572-9230</eissn><abstract>We consider the spectral gap of a uniformly chosen random
(
d
1
,
d
2
)
-biregular bipartite graph
G
with
|
V
1
|
=
n
,
|
V
2
|
=
m
, where
d
1
,
d
2
could possibly grow with
n
and
m
. Let
A
be the adjacency matrix of
G
. Under the assumption that
d
1
≥
d
2
and
d
2
=
O
(
n
2
/
3
)
,
we show that
λ
2
(
A
)
=
O
(
d
1
)
with high probability. As a corollary, combining the results from (Tikhomirov and Yousse in Ann Probab 47(1):362–419, 2019), we show that the second singular value of a uniform random
d
-regular digraph is
O
(
d
)
for
1
≤
d
≤
n
/
2
with high probability. Assuming
d
2
is fixed and
d
1
=
O
(
n
2
)
, we further prove that for a random
(
d
1
,
d
2
)
-biregular bipartite graph,
|
λ
i
2
(
A
)
-
d
1
|
=
O
(
d
1
)
for all
2
≤
i
≤
n
+
m
-
1
with high probability. The proofs of the two results are based on the size biased coupling method introduced in Cook et al. (Ann Probab 46(1):72–125, 2018) for random
d
-regular graphs and several new switching operations we define for random bipartite biregular graphs.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10959-022-01190-0</doi><tpages>35</tpages><orcidid>https://orcid.org/0000-0002-5665-0374</orcidid></addata></record> |
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| identifier | ISSN: 0894-9840 |
| ispartof | Journal of theoretical probability, 2023-06, Vol.36 (2), p.1269-1303 |
| issn | 0894-9840 1572-9230 |
| language | eng |
| recordid | cdi_proquest_journals_2817624141 |
| source | SpringerLink Journals |
| subjects | Eigenvalues Graph theory Graphs Mathematics Mathematics and Statistics Probability Theory and Stochastic Processes Statistics |
| title | On the Second Eigenvalue of Random Bipartite Biregular Graphs |
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