Some Indices of Alphabet Overlap Graph
The undirected de Bruijn graph is often used as the model of communication network for its useful properties, such as short diameter, small maximum vertex degree. In this paper, we consider the alphabet overlap graph G（k, d, s）： the vertex set V = {v｜v = （v1...vk）; vi ∈ {1,2,... ,d}, i = 1,2,... ,k}...
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| Published in | Journal of computer science and technology Vol. 27; no. 4; pp. 897 - 902 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Boston
Springer US
01.07.2012
Springer Nature B.V School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China%Normal School, Gansu Lianhe University, Lanzhou 730000, China |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1000-9000 1860-4749 1860-4749 |
| DOI | 10.1007/s11390-012-1261-9 |
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| Abstract | The undirected de Bruijn graph is often used as the model of communication network for its useful properties, such as short diameter, small maximum vertex degree. In this paper, we consider the alphabet overlap graph G（k, d, s）： the vertex set V = {v｜v = （v1...vk）; vi ∈ {1,2,... ,d}, i = 1,2,... ,k}; they are distinct and two vertices u = （ul...uk） and v = （vl... vk） are adjacent if and only if us＋i = vi or vs＋i = ui （i = 1,2,...,k - s）. In particular, when s = 1, G（k,d,s） is just an undirected de Bruijn graph. First, we give a formula to calculate the vertex degree of G（k, d, s）. Then, we use the corollary of Menger＇s theorem to prove that the connectivity of G（k, d, s） is 2d^s - 2d^2s-k for s ≥ k/2. |
|---|---|
| AbstractList | The undirected de Bruijn graph is often used as the model of communication network for its useful properties, such as short diameter, small maximum vertex degree. In this paper, we consider the alphabet overlap graph
G
(
k, d, s
): the vertex set
V
=
{v
|
v
= (
v
1
…v
k
);
v
i
∈ {1
,
2,…,
d
},
i
= 1
,
2,…,
k
}; they are distinct and two vertices
u
= (
u
1
…
u
k
) and
v
= (
v
1
…
v
k
) are adjacent if and only if
u
s
+
i
=
v
i
or
v
s
+
i
=
u
i
(
i
= 1, 2,…,
k − s
). In particular, when
s
= 1,
G
(
k, d, s
) is just an undirected de Bruijn graph. First, we give a formula to calculate the vertex degree of
G
(
k, d, s
). Then, we use the corollary of Menger’s theorem to prove that the connectivity of
G
(
k, d, s
) is 2
d
s
−
2
d
2
s−k
for
s ≥
k/
2. The undirected de Bruijn graph is often used as the model of communication network for its useful properties, such as short diameter, small maximum vertex degree. In this paper, we consider the alphabet overlap graph G(k, d, s): the vertex set V = {v|v = (v1…vk); vi ∈ {1, 2,…, d}, i = 1, 2,…, k}; they are distinct and two vertices u = (u1…uk) and v = (v1…vk) are adjacent if and only if us+i = vi or vs+i = ui (i = 1, 2,…, k − s). In particular, when s = 1, G(k, d, s) is just an undirected de Bruijn graph. First, we give a formula to calculate the vertex degree of G(k, d, s). Then, we use the corollary of Menger’s theorem to prove that the connectivity of G(k, d, s) is 2ds − 2d2s−k for s ≥ k/2. The undirected de Bruijn graph is often used as the model of communication network for its useful properties, such as short diameter, small maximum vertex degree. In this paper, we consider the alphabet overlap graph G（k, d, s）： the vertex set V = {v｜v = （v1...vk）; vi ∈ {1,2,... ,d}, i = 1,2,... ,k}; they are distinct and two vertices u = （ul...uk） and v = （vl... vk） are adjacent if and only if us＋i = vi or vs＋i = ui （i = 1,2,...,k - s）. In particular, when s = 1, G（k,d,s） is just an undirected de Bruijn graph. First, we give a formula to calculate the vertex degree of G（k, d, s）. Then, we use the corollary of Menger＇s theorem to prove that the connectivity of G（k, d, s） is 2d^s - 2d^2s-k for s ≥ k/2. The undirected de Bruijn graph is often used as the model of communication network for its useful properties, such as short diameter, small maximum vertex degree. In this paper, we consider the alphabet overlap graph G(k, d, s): the vertex set V={v|v=( v sub(1) ...v sub( )k; v sub( )i{1, 2,..., d}, i=1, 2,..., k}; they are distinct and two vertices u=(u sub(1)...u sub( )k and v=(v sub(1)...v sub( )k are adjacent if and only if u sub( )si v sub( )ior v sub( )si u sub( )i(i=1, 2,..., k-s). In particular, when s=1, G(k, d, s) is just an undirected de Bruijn graph. First, we give a formula to calculate the vertex degree of G(k, d, s). Then, we use the corollary of Menger's theorem to prove that the connectivity of G(k, d, s) is 2d super( )s-2d super(2)s-kfor s greater than or equal to k/2. |
| Author | 杨荣 杨兆兰 张和平 |
| AuthorAffiliation | School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China Normal School, Gansu Lianhe University, Lanzhou 730000, China |
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| Copyright | Springer Science+Business Media, LLC & Science Press, China 2012 Springer Science+Business Media, LLC & Science Press, China 2012. Copyright © Wanfang Data Co. Ltd. All Rights Reserved. |
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| Keywords | undirected de Bruijn graph vertex degree connectivity alphabet overlap graph |
| Language | English |
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| Notes | Rong Yang, Zhao-Lan Yang, He-Ping Zhang（1 School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China ;2Normal School, Gansu Lianhe University, Lanzhou 730000, China） 11-2296/TP undirected de Bruijn graph, alphabet overlap graph, vertex degree, connectivity The undirected de Bruijn graph is often used as the model of communication network for its useful properties, such as short diameter, small maximum vertex degree. In this paper, we consider the alphabet overlap graph G（k, d, s）： the vertex set V = {v｜v = （v1...vk）; vi ∈ {1,2,... ,d}, i = 1,2,... ,k}; they are distinct and two vertices u = （ul...uk） and v = （vl... vk） are adjacent if and only if us＋i = vi or vs＋i = ui （i = 1,2,...,k - s）. In particular, when s = 1, G（k,d,s） is just an undirected de Bruijn graph. First, we give a formula to calculate the vertex degree of G（k, d, s）. Then, we use the corollary of Menger＇s theorem to prove that the connectivity of G（k, d, s） is 2d^s - 2d^2s-k for s ≥ k/2. ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 ObjectType-Article-2 ObjectType-Feature-1 content type line 23 |
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| References_xml | – reference: LiuCHZhangKMSuper connectivity and restricted connectivity of undirected de Bruijn graphActa Mathematicae Applicatae Sinica.200225129351926718 – reference: RivalsERahmannSCombinatorics of periods in stringsJ. Combin. Theory, Series A200310419511320184221073.6870610.1016/S0097-3165(03)00123-7 – reference: WangSYuanJDNA computing of directed line-graphsCommunications Mathematical and in Computer Chemistry20065634794842312026 – reference: DebraKYaredNAttilaPOn the chromatic number of α-overlap graphsJournal of Combinatorial Mathematics and Computational Computing2010733131198.05060 – reference: Lin H. 1-Fair alternator designs for the de Bruijn network [Master’s Thesis]. National Sun Yat-sen University, 2006. – reference: Diestel R. Graph Theory (2nd edition), Springer-Verlag, 2000. – reference: Godbole A, Knisley D, Norwood R. Some properties of alphabet overlap graphs, http://arxiv.org/abs/math/0510094. – reference: Patwardhan R, Tang H, Kim S, Dalkilie M. An approximate de Bruijn graph approach to multiple local alignment and mitif discovery on protein sequences. In Proc. the 1st International Conference on Data Mining and Bioinformatics, Sept. 2006, pp.158–169. – reference: LiXYZhangHPEmbedding on alphabet overlap digraphsJ. Math. Chem.2010471627125766370567651710.1007/s10910-009-9530-8 – reference: AbedSMokhtariYAit-MohamedOTaharSNuMDG: A new tool for multiway decision graphs constructionJournal of Computer Science and Technology2011261139152278919710.1007/s11390-011-9421-x – reference: BlasewiczJHertzAKoblerDde WerraDOn some properties of DNA graphsDiscrete Appl. Math.1999981/2119172368310.1016/S0166-218X(99)00109-2 – volume: 98 start-page: 1 issue: 1/2 year: 1999 ident: 1261_CR7 publication-title: Discrete Appl. Math. doi: 10.1016/S0166-218X(99)00109-2 – ident: 1261_CR3 doi: 10.1007/11960669_14 – ident: 1261_CR4 doi: 10.1109/PDCAT.2006.118 – volume: 73 start-page: 3 year: 2010 ident: 1261_CR6 publication-title: Journal of Combinatorial Mathematics and Computational Computing – volume: 104 start-page: 95 issue: 1 year: 2003 ident: 1261_CR10 publication-title: J. Combin. Theory, Series A doi: 10.1016/S0097-3165(03)00123-7 – volume: 56 start-page: 479 issue: 3 year: 2006 ident: 1261_CR8 publication-title: Communications Mathematical and in Computer Chemistry – ident: 1261_CR2 – volume: 47 start-page: 62 issue: 1 year: 2010 ident: 1261_CR9 publication-title: J. Math. Chem. doi: 10.1007/s10910-009-9530-8 – ident: 1261_CR1 – volume: 26 start-page: 139 issue: 1 year: 2011 ident: 1261_CR5 publication-title: Journal of Computer Science and Technology doi: 10.1007/s11390-011-9421-x – volume: 25 start-page: 29 issue: 1 year: 2002 ident: 1261_CR11 publication-title: Acta Mathematicae Applicatae Sinica. |
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| SubjectTerms | Alphabets Apexes Artificial Intelligence Bruijn图 Communication networks Communications networks Computer Science Computer simulation Connectivity Data Structures and Information Theory Graph theory Graphs Information Systems Applications (incl.Internet) Mathematical analysis Mathematical models Short Paper Software Engineering Studies Theorems Theory of Computation Vertex sets 字母 连通性 通信网络 顶点度 顶点集 顶焦度 |
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| Title | Some Indices of Alphabet Overlap Graph |
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