Graph products are operations that combine two or more graphs to create a new graph, and these products can capture different structural relationships between the original graphs. There are several types of graph products, each with its own definition and properties.
The Cartesian product of two graphs \( G_1 = (V_1, E_1) \) and \( G_2 = (V_2, E_2) \) is a graph constructed by combining the vertices of the two graphs in a specific way.
In graph theory, the graph product is a way to combine two graphs to create a new graph. There are several types of graph products, each with different properties and applications.
The lexicographic product (or Cartesian product) of two graphs \( G = (V_G, E_G) \) and \( H = (V_H, E_H) \) is a graph denoted by \( G \cdot H \) (or sometimes \( G[H] \) or \( G \square H \)).
Min-plus matrix multiplication is an operation that extends the traditional concept of matrix multiplication using the minimum and plus operations instead of the standard addition and multiplication. Specifically, in min-plus algebra: 1. **Addition** in the min-plus context is defined as taking the minimum of two values. For example: \[ a \oplus b = \min(a, b) \] 2. **Multiplication** involves addition.
The modular product of graphs is a way to combine two graphs into a new one that captures certain structural properties of the original graphs. Specifically, it preserves the modularity of the vertex sets in each graph.
The rooted product of graphs is a specific operation that combines two graphs in a way that preserves some structural properties of both.
The strong product of two graphs \( G \) and \( H \), denoted as \( G \boxtimes H \), is a graph that combines the structures of both \( G \) and \( H \) in a way that incorporates features from both the Cartesian product and the tensor product of graphs.
The tensor product of graphs, also known as the Kronecker product or categorical product, is a way to combine two graphs into a new graph.
The zig-zag product is an operation on graphs, specifically useful in the field of combinatorial design and expander graphs. It allows the construction of a new graph from two existing graphs in a way that preserves certain properties, typically expanding size and connectivity characteristics. For two graphs \( G \) and \( H \): - Let \( G \) be a graph with vertex set \( V_G \) and \( H \) be a directed graph with vertex set \( V_H \).
Articles by others on the same topic
There are currently no matching articles.