Bouquet graph

In mathematics, a bouquet graph $$B_m$$, for an integer parameter $$m$$, is an undirected graph with one vertex and $$m$$ edges, all of which are self-loops. It is the graph-theoretic analogue of the topological rose, 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.

Although bouquets have a very simple structure as graphs, they are of some importance in topological graph theory because their graph embeddings 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 of the graph, or alternatively by contracting the edges of any spanning tree.

In graph-theoretic approaches to group theory, every Cayley–Serre graph (a variant of Cayley graphs with doubled edges) can be represented as the covering graph of a bouquet.