Layer structure of De Bruijn and Kautz digraphs. An application to deflection routing

In the main part of this paper we present polynomial expressions for the cardinalities of some sets of interest of the nice distance-layer structure of the well-known De Bruijn and Kautz digraphs. More precisely, given a vertex v, let Si⋆ (v) be the set of vertices at distance i from v. We show that...

Celý popis

Uloženo v:

Podrobná bibliografie
Vydáno v:	Electronic notes in discrete mathematics Ročník 54; s. 157 - 162
Hlavní autoři:	Fàbrega, J., Martí-Farré, J., Muñoz, X.
Médium:	Journal Article Publikace
Jazyk:	angličtina
Vydáno:	Elsevier B.V 01.10.2016
Témata:	11 Number theory 11C Polynomials and matrices 12 Field theory and polynomials 12Y05 Computational aspects of field theory and polynomials Classificació AMS De Bruijn and Kautz digraphs Deflection routing General iterated line digraphs Matemàtiques i estadística Matrices Matrius (Matemàtica) Polinomis Polynomials Teoria de cossos i polinomis Teoria de nombres Àlgebra Àrees temàtiques de la UPC General iterated line digraphs Deflection routing De Bruijn and Kautz digraphs
ISSN:	1571-0653, 1571-0653
On-line přístup:	Získat plný text
Tagy:	Přidat tag Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!

Abstract	In the main part of this paper we present polynomial expressions for the cardinalities of some sets of interest of the nice distance-layer structure of the well-known De Bruijn and Kautz digraphs. More precisely, given a vertex v, let Si⋆ (v) be the set of vertices at distance i from v. We show that \|Si⋆(v)\|=di−ai−1di−1−⋯−a1d−a0, where d is the degree of the digraph and the coefficients ak∈{0,1} are explicitly calculated. Analogously, let w be a vertex adjacent from v such that Si⋆(v)∩Sj⁎(w)≠∅ for some j. We prove that \|Si⋆(v)∩Sj⁎(w)\|=di−bi−1di−1−…−b1d−b0, where the coefficients bt∈{0,1} are determined from the coefficients ak of the polynomial expression of \|Si⋆(v)\|. An application to deflection routing in De Bruijn and Kautz networks serves as motivation for our study. It is worth-mentioning that our analysis can be extended to other families of digraphs on alphabet or to general iterated line digraphs.
AbstractList	In the main part of this paper we present polynomial expressions for the cardinalities of some sets of interest of the nice distance-layer structure of the well-known De Bruijn and Kautz digraphs. More precisely, given a vertex $v$, let $S_{i}^star(v)$ be the set of vertices at distance $i$ from $v$. We show that $\|S_{i}^star(v)\|=d^i-a_{i-1}d^{i-1}-cdots -a_{1} d-a_{0}$, where $d$ is the degree of the digraph and the coefficients $a_{k}in{0,1}$ are explicitly calculated. Analogously, let $w$ be a vertex adjacent from $v$ such that $S_{i}^star(v)cap S_j^{ast}(w)neq emptyset$ for some $j$. We prove that $big \|S_{i}^star(v) cap S_j^{ast}(w) big \|=d^i-b_{i-1}d^{i-1}-ldots -b_{1} d-b_{0},$ where the coefficients $b_{t}in{0,1}$ are determined from the coefficients $a_k$ of the polynomial expression of $\|S_{i}^star(v)\|$. An application to deflection routing in De Bruijn and Kautz networks serves as motivation for our study. It is worth-mentioning that our analysis can be extended to other families of digraphs on alphabet or to general iterated line digraphs. Peer Reviewed In the main part of this paper we present polynomial expressions for the cardinalities of some sets of interest of the nice distance-layer structure of the well-known De Bruijn and Kautz digraphs. More precisely, given a vertex v, let Si⋆ (v) be the set of vertices at distance i from v. We show that \|Si⋆(v)\|=di−ai−1di−1−⋯−a1d−a0, where d is the degree of the digraph and the coefficients ak∈{0,1} are explicitly calculated. Analogously, let w be a vertex adjacent from v such that Si⋆(v)∩Sj⁎(w)≠∅ for some j. We prove that \|Si⋆(v)∩Sj⁎(w)\|=di−bi−1di−1−…−b1d−b0, where the coefficients bt∈{0,1} are determined from the coefficients ak of the polynomial expression of \|Si⋆(v)\|. An application to deflection routing in De Bruijn and Kautz networks serves as motivation for our study. It is worth-mentioning that our analysis can be extended to other families of digraphs on alphabet or to general iterated line digraphs.
Author	Muñoz, X. Martí-Farré, J. Fàbrega, J.
Author_xml	– sequence: 1 givenname: J. surname: Fàbrega fullname: Fàbrega, J. email: josep.fabrega@upc.edu – sequence: 2 givenname: J. surname: Martí-Farré fullname: Martí-Farré, J. email: jaume.marti@upc.edu – sequence: 3 givenname: X. surname: Muñoz fullname: Muñoz, X. email: xavier.munoz@upc.edu
BookMark	eNp9kMtKAzEUhoNUsFVfwFVeoGOSuWQG3NR6xYIbuw6Z5KRmaGdKkhHq05tpC4qLLg7nAt_h55ugUdu1gNANJQkltLhtEmj1JmFxTkiVEFaeoTHNOZ2SIk9Hf-YLNPG-ISQtKc_HaLmQO3DYB9er0DvAncEPgO9db5sWy1bjN9mHb6ztysntp0_wLJ6327VVMtiuxaHDGswa1H5zXR9su7pC50auPVwf-yVaPj1-zF-mi_fn1_lsMVUs52GqpDE6L0mhq4rxjGU1Z7Koa1brtEiBZll8bQrNGasMJwBlVaaVNiCJkozJ9BLRw1_leyUcKHAxluik_V2GYoQzwUpWFjwy7Mi4znsHRmyd3Ui3E5SIQaZoxCBTDDIFqUSUGaHyH6Rs2AsITtr1afTugEIU8WXBCa8stAq0jRmD0J09hf8AitmTTg
CitedBy_id	crossref_primary_10_1002_net_22177
Cites_doi	10.1109/TC.1984.1676455 10.1016/S0166-218X(98)00120-6 10.1109/TCYB.2014.2360680 10.1016/0166-218X(92)90135-W 10.14257/ijfgcn.2015.8.1.23 10.1109/TCOM.1964.1088883
ContentType	Journal Article Publication
Contributor	Universitat Politècnica de Catalunya. COMBGRAPH - Combinatòria, Teoria de Grafs i Aplicacions Universitat Politècnica de Catalunya. Departament de Matemàtiques
Contributor_xml	– sequence: 1 fullname: Universitat Politècnica de Catalunya. Departament de Matemàtiques – sequence: 2 fullname: Universitat Politècnica de Catalunya. COMBGRAPH - Combinatòria, Teoria de Grafs i Aplicacions
Copyright	2016 Elsevier B.V. Attribution-NonCommercial-NoDerivs 3.0 Spain info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by-nc-nd/3.0/es
Copyright_xml	– notice: 2016 Elsevier B.V. – notice: Attribution-NonCommercial-NoDerivs 3.0 Spain info:eu-repo/semantics/openAccess <a href="http://creativecommons.org/licenses/by-nc-nd/3.0/es/">http://creativecommons.org/licenses/by-nc-nd/3.0/es/</a>
DBID	AAYXX CITATION XX2
DOI	10.1016/j.endm.2016.09.028
DatabaseName	CrossRef Recercat
DatabaseTitle	CrossRef
DatabaseTitleList
DeliveryMethod	fulltext_linktorsrc
Discipline	Computer Science
EISSN	1571-0653
EndPage	162
ExternalDocumentID	oai_recercat_cat_2072_282867 10_1016_j_endm_2016_09_028 S1571065316301226
GrantInformation_xml	– fundername: Ministerio de Economía y Competitividad grantid: MTM2014-60127-P funderid: http://dx.doi.org/10.13039/501100003329 – fundername: Catalan Research Council grantid: 2014-SGR-1147
GroupedDBID	--K --M .~1 0R~ 1B1 1~. 1~5 4.4 457 4G. 5GY 5VS 7-5 71M 8P~ AACTN AAEDT AAEDW AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAQXK AASFE AAXUO ABAOU ABJNI ABMAC ABXDB ABYKQ ACAZW ACDAQ ACGFO ACGFS ACNNM ACRLP ACXMD ADBBV ADEZE ADGUI ADMUD AEBSH AEKER AFKWA AFTJW AGHFR AGUBO AGYEJ AIEXJ AIGVJ AIKHN AITUG AJMQA AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ AOUOD APLSM ARUGR ASPBG AVWKF AXJTR AZFZN BKOJK BLXMC CS3 EBS EFJIC EFLBG EJD EO8 EO9 EP2 EP3 FDB FEDTE FGOYB FIRID FNPLU FYGXN G-Q GBLVA GBOLZ HAMUX HVGLF HZ~ IHE J1W KOM M41 MHUIS MO0 M~E N9A O-L O9- OAUVE OZT P-8 P-9 P2P PC. Q38 R2- RIG ROL RPZ SDF SDG SES SEW SPC SPCBC SSB SSD SSV SSW SSZ T5K ~G- 9DU AATTM AAXKI AAYWO AAYXX ABWVN ACLOT ACRPL ACVFH ADCNI ADNMO AEIPS AEUPX AFPUW AGQPQ AIGII AIIUN AKBMS AKRWK AKYEP ANKPU CITATION EFKBS ~HD XX2
ID	FETCH-LOGICAL-c257t-caffd5806d9927424b72a6bb2bd363e144deff6d7229f70ee89839dfea0ca22a3
ISSN	1571-0653
IngestDate	Fri Nov 07 13:53:54 EST 2025 Sat Nov 29 01:47:08 EST 2025 Tue Nov 18 22:41:10 EST 2025 Fri Feb 23 02:23:31 EST 2024
IsDoiOpenAccess	true
IsOpenAccess	true
IsPeerReviewed	false
IsScholarly	true
Keywords	General iterated line digraphs Deflection routing De Bruijn and Kautz digraphs
Language	English
LinkModel	OpenURL
MergedId	FETCHMERGED-LOGICAL-c257t-caffd5806d9927424b72a6bb2bd363e144deff6d7229f70ee89839dfea0ca22a3
OpenAccessLink	https://recercat.cat/handle/2072/282867
PageCount	6
ParticipantIDs	csuc_recercat_oai_recercat_cat_2072_282867 crossref_primary_10_1016_j_endm_2016_09_028 crossref_citationtrail_10_1016_j_endm_2016_09_028 elsevier_sciencedirect_doi_10_1016_j_endm_2016_09_028
PublicationCentury	2000
PublicationDate	2016-10-01
PublicationDateYYYYMMDD	2016-10-01
PublicationDate_xml	– month: 10 year: 2016 text: 2016-10-01 day: 01
PublicationDecade	2010
PublicationTitle	Electronic notes in discrete mathematics
PublicationYear	2016
Publisher	Elsevier B.V
Publisher_xml	– name: Elsevier B.V
References	Haeri, Trajkovic (br0070) 2015; 45 Zheng, Hu, Luo, Wu (br0080) 2015; 8 Baran (br0010) 1964; 12 Bermond, Peyrat (br0020) 1989 Gómez, Fiol, Yebra (br0040) 1992; 37/38 Fàbrega, Muñoz (br0050) 2003; vol. 2790 Gómez, Padró, Perennes (br0030) 1998; 89 Fiol, Yebra, Alegre (br0060) 1984; C-33 Fàbrega (10.1016/j.endm.2016.09.028_br0050) 2003; vol. 2790 Bermond (10.1016/j.endm.2016.09.028_br0020) 1989 Gómez (10.1016/j.endm.2016.09.028_br0030) 1998; 89 Fiol (10.1016/j.endm.2016.09.028_br0060) 1984; C-33 Haeri (10.1016/j.endm.2016.09.028_br0070) 2015; 45 Baran (10.1016/j.endm.2016.09.028_br0010) 1964; 12 Gómez (10.1016/j.endm.2016.09.028_br0040) 1992; 37/38 Zheng (10.1016/j.endm.2016.09.028_br0080) 2015; 8
References_xml	– volume: 89 start-page: 107 year: 1998 end-page: 123 ident: br0030 article-title: Large Generalized Cycles publication-title: Discrete Appl. Math. – start-page: 279 year: 1989 end-page: 294 ident: br0020 article-title: De Bruijn and Kautz networks: A competitor for the hypercube? publication-title: Hypercube and Distributed Computers – volume: 12 start-page: 1 year: 1964 end-page: 9 ident: br0010 article-title: On distributed communications networks publication-title: IEEE Trans. Comm. Sys. – volume: 37/38 start-page: 227 year: 1992 end-page: 243 ident: br0040 article-title: Graphs on alphabets as models for large interconnection networks publication-title: Discrete Appl. Math. – volume: vol. 2790 start-page: 989 year: 2003 end-page: 994 ident: br0050 article-title: A study of network capacity under deflection routing schemes publication-title: Euro-Par 2003 Parallel Processing – volume: 45 start-page: 316 year: 2015 end-page: 327 ident: br0070 article-title: Intelligent deflection routing in buffer-less networks publication-title: IEEE Transactions on Cybernetics – volume: C-33 start-page: 400 year: 1984 end-page: 403 ident: br0060 article-title: Line digraph iterations and the (d, k) digraph problem publication-title: IEEE Trans. Comput. – volume: 8 start-page: 227 year: 2015 end-page: 236 ident: br0080 article-title: Study of Deflection Routing from an Information-theoretic Perspective publication-title: International Journal of Future Generation Communication and Networking – volume: C-33 start-page: 400 year: 1984 ident: 10.1016/j.endm.2016.09.028_br0060 article-title: Line digraph iterations and the (d, k) digraph problem publication-title: IEEE Trans. Comput. doi: 10.1109/TC.1984.1676455 – volume: 89 start-page: 107 year: 1998 ident: 10.1016/j.endm.2016.09.028_br0030 article-title: Large Generalized Cycles publication-title: Discrete Appl. Math. doi: 10.1016/S0166-218X(98)00120-6 – volume: 45 start-page: 316 issue: 2 year: 2015 ident: 10.1016/j.endm.2016.09.028_br0070 article-title: Intelligent deflection routing in buffer-less networks publication-title: IEEE Transactions on Cybernetics doi: 10.1109/TCYB.2014.2360680 – volume: 37/38 start-page: 227 year: 1992 ident: 10.1016/j.endm.2016.09.028_br0040 article-title: Graphs on alphabets as models for large interconnection networks publication-title: Discrete Appl. Math. doi: 10.1016/0166-218X(92)90135-W – volume: 8 start-page: 227 issue: 1 year: 2015 ident: 10.1016/j.endm.2016.09.028_br0080 article-title: Study of Deflection Routing from an Information-theoretic Perspective publication-title: International Journal of Future Generation Communication and Networking doi: 10.14257/ijfgcn.2015.8.1.23 – start-page: 279 year: 1989 ident: 10.1016/j.endm.2016.09.028_br0020 article-title: De Bruijn and Kautz networks: A competitor for the hypercube? – volume: 12 start-page: 1 year: 1964 ident: 10.1016/j.endm.2016.09.028_br0010 article-title: On distributed communications networks publication-title: IEEE Trans. Comm. Sys. doi: 10.1109/TCOM.1964.1088883 – volume: vol. 2790 start-page: 989 year: 2003 ident: 10.1016/j.endm.2016.09.028_br0050 article-title: A study of network capacity under deflection routing schemes
SSID	ssj0038175
Score	2.0300531
Snippet	In the main part of this paper we present polynomial expressions for the cardinalities of some sets of interest of the nice distance-layer structure of the...
SourceID	csuc crossref elsevier
SourceType	Open Access Repository Enrichment Source Index Database Publisher
StartPage	157
SubjectTerms	11 Number theory 11C Polynomials and matrices 12 Field theory and polynomials 12Y05 Computational aspects of field theory and polynomials Classificació AMS De Bruijn and Kautz digraphs Deflection routing General iterated line digraphs Matemàtiques i estadística Matrices Matrius (Matemàtica) Polinomis Polynomials Teoria de cossos i polinomis Teoria de nombres Àlgebra Àrees temàtiques de la UPC
Title	Layer structure of De Bruijn and Kautz digraphs. An application to deflection routing
URI	https://dx.doi.org/10.1016/j.endm.2016.09.028 https://recercat.cat/handle/2072/282867
Volume	54
hasFullText	1
inHoldings	1
isFullTextHit
isPrint
journalDatabaseRights	– providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 customDbUrl: eissn: 1571-0653 dateEnd: 20190331 omitProxy: false ssIdentifier: ssj0038175 issn: 1571-0653 databaseCode: AIEXJ dateStart: 19990301 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier – providerCode: PRVHPJ databaseName: ROAD: Directory of Open Access Scholarly Resources customDbUrl: eissn: 1571-0653 dateEnd: 20191231 omitProxy: false ssIdentifier: ssj0038175 issn: 1571-0653 databaseCode: M~E dateStart: 19990101 isFulltext: true titleUrlDefault: https://road.issn.org providerName: ISSN International Centre
link	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1Lj9MwELZK4cCFN6K85AMnqlSx83ByXOgiBMuKwy7qLXJiR2pVklWbrBYO_A5-LjO2k3QLrNgDh0atm0zTzOfxeDzzmZBXJQt1yFTo5SoKvbBMA-hzAfdymWBCZZ5zU8f95UgcHyeLRfp5NPrZ1cKcr0VVJRcX6dl_VTW0gbKxdPYa6u6FQgO8B6XDEdQOx39S_JH8hgzNhhcWVwfAG5xjDkC7XNnE44-ybb5P1dJwVW9nGBfcWcZGZ1Tpcq3tHuKbum260a0L4A8751R1o21C7RLMD_jf0689DexQWGIW432Ye9sg7ofZEAbfNObLuYdkkHbRfu-UFlvfsNoEuhez3SgFi_t8t96wCsywssTAM_2HNmeNo3DHnDJLXu1GZmbt9m9G38YfVjNdKeQWYLFhrnU155cYtvdGvj4fsUt1W2UoI0MZmZ9mIOMGuclFlKK9_PTjsBvjkdYwMky87h-4ciybObh_H5dcnnGxbYsdz2fHmzm5R-64aQg9sPC5T0a6ekDudlt8UGfxH5JTgybao4nWJZ1ratFEAU3UoIn2aKIH0DygiTY1HdBEHZoekdN3hydv33tuJw6vAJPeeIUsSxUlfqzSFNf2w1xwGUNHzlUQBxom5SCqjJXgPC2Fr3WSguOtSi39QnIug8dkXNWVfkJowViRaBUJGQdhUSYS2ZgYVwxcVSVkNCGse1hZ4WjqcbeUdfZ3JU3ItL_mzJK0XHn2a9RBBh6F3sDDyJBhvf-AL-4LnmEsIhYTEnWaypxPan3NDNB3xY88vdYtPSO3h07znIxBqfoFuVWcN8vt5qWB3i-sYalT
linkProvider	ISSN International Centre
openUrl	ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Layer+structure+of+De+Bruijn+and+Kautz+digraphs.+An+application+to+deflection+routing&rft.jtitle=Electronic+notes+in+discrete+mathematics&rft.au=F%C3%A0brega%2C+J.&rft.au=Mart%C3%AD-Farr%C3%A9%2C+J.&rft.au=Mu%C3%B1oz%2C+X.&rft.date=2016-10-01&rft.issn=1571-0653&rft.eissn=1571-0653&rft.volume=54&rft.spage=157&rft.epage=162&rft_id=info:doi/10.1016%2Fj.endm.2016.09.028&rft.externalDBID=n%2Fa&rft.externalDocID=10_1016_j_endm_2016_09_028
thumbnail_l	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1571-0653&client=summon
thumbnail_m	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1571-0653&client=summon
thumbnail_s	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1571-0653&client=summon