In mathematics, a bouquet graph

B_m

, for an integer parameter

m

, is an

undirected graph In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' v ...

with one

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

and

m

edges, all of which are

self-loop In graph theory, a loop (also called a self-loop or a ''buckle'') is an edge (graph theory), edge that connects a vertex (graph theory), vertex to itself. A Graph (discrete mathematics)#Simple graph, simple graph contains no loops. Depending on ...

s. It is the

graph-theoretic In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

analogue of the topological bouquet, a space of

m

circles joined at a point. When the context of graph theory is clear, it can be called more simply a bouquet. Self-trial ribbon graph

Although bouquets have a very simple structure as graphs, they are of some importance in

topological graph theory In mathematics, topological graph theory is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces. It also studies immersions of graphs. Embedding a graph in ...

because their

graph embedding In topological graph theory, an embedding (also spelled imbedding) of a Graph (discrete mathematics), graph G on a surface (mathematics), surface \Sigma is a representation of G on \Sigma in which points of \Sigma are associated with graph the ...

s can still be non-trivial. In particular, every cellularly embedded graph can be reduced to an embedded bouquet by a partial duality applied to the edges of any

spanning tree In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is not ...

of the graph, or alternatively by contracting the edges of any spanning tree. In graph-theoretic approaches to

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, every Cayley–Serre graph (a variant of Cayley graphs with doubled edges) can be represented as the

covering graph In the mathematical discipline of graph theory, a graph is a covering graph of another graph if there is a covering map from the vertex set of to the vertex set of . A covering map is a surjection and a local isomorphism: the neighbourhood of ...

of a bouquet.

References

{{reflist, refs= {{citation , last1 = Beineke , first1 = Lowell W. , author1-link = L. W. Beineke , last2 = Wilson , first2 = Robin J. , author2-link = Robin Wilson (mathematician) , doi = 10.1017/CBO9781139087223 , isbn = 978-0-521-80230-7 , mr = 2581536 , page = 5 , publisher = Cambridge University Press, Cambridge , series = Encyclopedia of Mathematics and its Applications , title = Topics in topological graph theory , volume = 128 , year = 2009 {{citation , last1 = Ellis-Monaghan , first1 = Joanna A. , author1-link = Jo Ellis-Monaghan , last2 = Moffatt , first2 = Iain , issue = 3 , journal =

Transactions of the American Mathematical Society The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 p ...

, mr = 2869185 , pages = 1529–1569 , title = Twisted duality for embedded graphs , doi = 10.1090/S0002-9947-2011-05529-7 , volume = 364 , year = 2012, arxiv = 0906.5557 {{citation , last = Sunada , first = Toshikazu , doi = 10.1007/978-4-431-54177-6 , isbn = 978-4-431-54176-9 , mr = 3014418 , page = 69 , publisher = Springer, Tokyo , series = Surveys and Tutorials in the Applied Mathematical Sciences , title = Topological Crystallography: With a View Towards Discrete Geometric Analysis , url = https://books.google.com/books?id=6cNEAAAAQBAJ&pg=PA69 , volume = 6 , year = 2013 Parametric families of graphs