Antipodal graphs and digraphs
The antipodal graph of a graph G, denoted by A(G), has the same vertex set as G with an edge joining vertices u and v if d(u,v) is equal to the diameter of G. (If G is disconnected, then diam G=∞.) This definition is extended to a digraph D where the arc (u,v) is included in A(D) if d(u,v) is the di...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Wiley
1993-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171293000717 |
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| _version_ | 1849402269835984896 |
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| author | Garry Johns Karen Sleno |
| author_facet | Garry Johns Karen Sleno |
| author_sort | Garry Johns |
| collection | DOAJ |
| description | The antipodal graph of a graph G, denoted by A(G), has the same vertex set
as G with an edge joining vertices u and v if d(u,v) is equal to the diameter of G. (If G is
disconnected, then diam G=∞.) This definition is extended to a digraph D where the arc
(u,v) is included in A(D) if d(u,v) is the diameter of D. It is shown that a digraph D is an
antipodal digraph if and only if D is the antipodal digraph of its complement. This generalizes
a known characterization for antipodal graphs and provides an improved proof. Examples
and properties of antipodal digraphs are given. A digraph D is self-antipodal if A(D) is
isomorphic to D. Several characteristics of a self-antipodal digraph D are given including
sharp upper and lower bounds on the size of D. Similar results are given for self-antipodal
graphs. |
| format | Article |
| id | doaj-art-8b68a79ed7ae4e35bff4154fd3337ab8 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1993-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-8b68a79ed7ae4e35bff4154fd3337ab82025-08-20T03:37:34ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251993-01-0116357958610.1155/S0161171293000717Antipodal graphs and digraphsGarry Johns0Karen Sleno1Department of Mathematical Sciences, Saginaw Valley State University, University Center 48710, Michigan, USADepartment of Mathematical Sciences, Saginaw Valley State University, University Center 48710, Michigan, USAThe antipodal graph of a graph G, denoted by A(G), has the same vertex set as G with an edge joining vertices u and v if d(u,v) is equal to the diameter of G. (If G is disconnected, then diam G=∞.) This definition is extended to a digraph D where the arc (u,v) is included in A(D) if d(u,v) is the diameter of D. It is shown that a digraph D is an antipodal digraph if and only if D is the antipodal digraph of its complement. This generalizes a known characterization for antipodal graphs and provides an improved proof. Examples and properties of antipodal digraphs are given. A digraph D is self-antipodal if A(D) is isomorphic to D. Several characteristics of a self-antipodal digraph D are given including sharp upper and lower bounds on the size of D. Similar results are given for self-antipodal graphs.http://dx.doi.org/10.1155/S0161171293000717antipodal graphsantipodal digraphs. |
| spellingShingle | Garry Johns Karen Sleno Antipodal graphs and digraphs International Journal of Mathematics and Mathematical Sciences antipodal graphs antipodal digraphs. |
| title | Antipodal graphs and digraphs |
| title_full | Antipodal graphs and digraphs |
| title_fullStr | Antipodal graphs and digraphs |
| title_full_unstemmed | Antipodal graphs and digraphs |
| title_short | Antipodal graphs and digraphs |
| title_sort | antipodal graphs and digraphs |
| topic | antipodal graphs antipodal digraphs. |
| url | http://dx.doi.org/10.1155/S0161171293000717 |
| work_keys_str_mv | AT garryjohns antipodalgraphsanddigraphs AT karensleno antipodalgraphsanddigraphs |