A Lotus graph, most commonly known as a friendship graph or flower graph, is a specific type of graph in graph theory characterized by a collection of triangles (3-cycles) sharing a single common vertex. While this is its widely recognized form, specialized variations, such as the "Stem-Lotus graph," also exist within particular research contexts.
The Core Concept: Lotus Graph (Friendship Graph)
The most prevalent understanding of a Lotus graph, often denoted as $F_k$ or sometimes $L_k$, refers to a graph constructed by taking $k$ copies of the cycle graph $C_3$ (a triangle) and joining them at a single common vertex. This central vertex serves as the "apex" or "hub" for all the triangles, giving it a floral appearance.
- Structure: Each triangle shares one vertex with all other triangles, while its other two vertices are unique to that specific triangle.
- Key Properties:
- Vertices: A Lotus graph $F_k$ has a total of $2k + 1$ vertices. This includes one central vertex and $2k$ peripheral vertices (two for each of the $k$ triangles).
- Edges: It contains $3k$ edges. Each triangle contributes three edges, and since they only share a vertex (not edges), all edges are distinct.
- Cycles: It consists of $k$ distinct cycles of length 3 (triangles).
- Connectivity: It is a connected graph.
Visualizing a Lotus Graph
Imagine a daisy or a flower with multiple petals. The center of the flower is the common vertex, and each petal represents a triangle.
Table: Properties of Friendship Graphs (Lotus Graphs)
| Property | Description | Formula (for $F_k$) |
|---|---|---|
| Vertices | Total number of nodes in the graph | $2k + 1$ |
| Edges | Total number of connections | $3k$ |
| Components | Number of triangles sharing the central vertex | $k$ |
| Diameter | The maximum shortest distance between any two vertices | 2 |
Examples of Lotus Graphs
- $F_1$ (A Single Triangle): The simplest form, though typically $k \ge 2$ is used to emphasize the "friendship" or "flower" aspect. It has 3 vertices and 3 edges.
- $F_2$ (Bow-tie Graph): Two triangles sharing a common vertex. It has 5 vertices and 6 edges.
- $F_3$: Three triangles sharing a common vertex. It has 7 vertices and 9 edges.
Applications and Significance
Lotus graphs, due to their simple yet distinct structure, are often used as:
- Benchmarks in algorithm testing: Their predictable properties make them useful for testing graph algorithms.
- Models for networks: They can represent simplified star-like networks where multiple connections converge at a central point.
- Educational tools: They serve as excellent examples for illustrating basic graph theory concepts like cycles, connectivity, and degree sequences.
Specialized Concept: The Stem-Lotus Graph
Beyond the commonly recognized Lotus graph (friendship graph), there exists a more specialized and complex construction known as a Stem-Lotus graph. This term is typically found in specific research papers and defines a graph with a unique method of formation, rather than the simple union of triangles.
A Stem-Lotus graph is a graph obtained from a shell graph $C(2n+3, 2n)$ where $n \geq 1$ by adding a vertex in between each pair of adjacent vertices on the cycle, adding an edge in apex and two more chords.
Deconstructing the Stem-Lotus Graph Construction
Let's break down the elements involved in creating a Stem-Lotus graph:
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Starting Point: A Shell Graph $C(2n+3, 2n)$
- A shell graph is generally a cycle graph with additional chords sharing a common endpoint (an "apex"). The notation $C(2n+3, 2n)$ describes a base structure that begins with a cycle of $2n+3$ vertices and incorporates $2n$ additional edges, usually as chords connected to an apex vertex, forming a dense core. The parameter $n$ must be an integer greater than or equal to 1.
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Subdivision of the Cycle:
- For every edge that forms part of the primary cycle of the initial shell graph, a new vertex is inserted. This process, known as subdivision, effectively doubles the length of the cycle and increases the total number of vertices involved in the cycle's path. If the original cycle had $2n+3$ edges, it will now have $2(2n+3)$ edges and $2(2n+3)$ vertices along its path.
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Apex Edge Addition:
- An additional edge is introduced involving the "apex" of the original shell graph. This likely connects the central apex vertex to one of the newly added subdivision vertices or to another existing vertex in a specific manner, further modifying the graph's connectivity.
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Two More Chords:
- Finally, two extra edges, referred to as "chords," are added to the graph. These chords bypass parts of the existing cycle or connect non-adjacent vertices, making the graph even denser and more complex.
This intricate construction indicates that a Stem-Lotus graph is a highly structured graph with specific properties, often studied in specialized areas of graph theory focusing on various graph parameters.
Distinguishing Lotus Graphs and Stem-Lotus Graphs
It's crucial to understand that the term "Lotus graph" most commonly refers to the simple and elegant friendship graph ($F_k$), composed of triangles sharing a central vertex. The "Stem-Lotus graph," on the other hand, is a much more intricate construction derived from a specialized shell graph through a series of specific modifications. While both terms include "Lotus," they describe distinct graph structures arising in different contexts within graph theory research.
