Glossary

12.3 Comparing Graphs

4 min read•june 18, 2024

Graph is a key concept in , focusing on structural equivalence between graphs. It's crucial for understanding relationships in various fields, from social networks to chemical compounds.

Techniques for determining isomorphism include analyzing degrees, cyclic subgraphs, and graph invariants. Identifying isomorphisms involves mapping vertices while preserving . Complementary graphs offer another tool for comparison in this fascinating area of study.

Graph Isomorphism

Isomorphism determination techniques

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  • Analyze the structure, vertex degrees, and cyclic subgraphs of two graphs to determine if they are
    • graphs have the same number of vertices, edges, and overall structure despite potentially different vertex labels or visual representations
    • Corresponding vertices in isomorphic graphs have the same , which is the number of edges incident to each vertex
  • Compare the degree sequences (lists of vertex degrees) of the two graphs
    • If the degree sequences differ, the graphs are not isomorphic as they have different vertex degree distributions
  • Identify and count the presence of cyclic subgraphs such as triangles or squares in each graph
    • If the number or types of cyclic subgraphs are not the same, the graphs are not isomorphic as they have different substructures
  • Graphs that have the same number of vertices, edges, , and cyclic subgraphs are likely to be isomorphic
  • Consider graph invariants, which are properties that remain unchanged under isomorphism, to further compare graphs

Identification of graph isomorphisms

  • An isomorphism is a ff that maps vertices of one graph G1G_1 to vertices of another graph G2G_2 while preserving adjacency relationships
    • If vertices uu and vv are connected by an in G1G_1, then their mapped vertices f(u)f(u) and f(v)f(v) must also be connected by an in G2G_2
  • To identify an isomorphism between two graphs, establish a between their vertices
    • Each vertex in G1G_1 should be paired with a unique vertex in G2G_2 such that the edge connections between corresponding vertices are preserved
  • Name the isomorphism by listing the vertex correspondences as ordered pairs
    • For example, f={(a,1),(b,2),(c,3),(d,4)}f = \{(a, 1), (b, 2), (c, 3), (d, 4)\} indicates that vertex aa in G1G_1 maps to vertex 11 in G2G_2, bb maps to 22, cc maps to 33, and dd maps to 44
  • Graph automorphisms are isomorphisms from a graph to itself, revealing symmetries within the graph structure

Complementary graphs for comparison

  • The of a graph GG, denoted as Gˉ\bar{G}, is a graph with the same as GG but with edges connecting vertices that are not adjacent in GG
    • If vertices uu and vv are connected by an edge in GG, they are not connected by an edge in Gˉ\bar{G}
    • Conversely, if vertices uu and vv are not connected by an edge in GG, they are connected by an edge in Gˉ\bar{G}
  • Constructing the complements of two graphs can aid in determining their isomorphism
    • If G1G_1 is isomorphic to G2G_2, then their complements G1ˉ\bar{G_1} and G2ˉ\bar{G_2} are also isomorphic
  • Analyze the structure, vertex degrees, and cyclic subgraphs of the complementary graphs to compare their properties
    • If the complements have the same structure, degree sequence, and cyclic subgraphs, the original graphs are likely to be isomorphic

Graph Theory Concepts

  • Graph theory provides a framework for studying and comparing graphs
  • Adjacency matrices can be used to represent graphs and aid in isomorphism comparisons
  • measures how well-connected a graph is and can be used to compare graph structures
  • Various algorithms in graph theory can be applied to determine isomorphism and analyze graph properties

Applications and Examples

  • Isomorphic graphs can represent the same underlying structure in various contexts, such as:
    • Social networks with different individuals but identical connection patterns (Facebook, Twitter)
    • Chemical compounds with different atom labels but the same bonding structure (isomers)
  • Examples of non-isomorphic graphs include:
    • A , which is a of length 3, and a , which is a of length 4
    • A , where all vertices are connected to each other, and a , where one central vertex is connected to all other vertices
  • Complementary graphs can reveal insights into graph properties and relationships, for instance:
    • The of a complete graph is an empty graph with no edges
    • The complement of a , where vertices are divided into two sets with no edges within each set, is also a bipartite graph

Key Terms to Review (33)

Adjacency: Adjacency refers to the relationship between two vertices in a graph when they are directly connected by an edge. This concept is crucial in understanding the structure and properties of graphs, as it defines how vertices interact and relate to each other. Adjacency can impact various features of a graph, such as connectivity, pathfinding, and network flow, playing a significant role in more complex topics like cycles and graph comparisons.
Adjacency matrix: An adjacency matrix is a square grid used to represent a finite graph, where each cell indicates whether pairs of vertices are adjacent or not in the graph. This matrix provides a way to store and analyze graph structures efficiently, making it easier to compare different graphs and investigate properties like connectivity and pathfinding. In particular, it plays a crucial role in algorithms used to determine Euler circuits, as it simplifies the identification of edges connecting vertices.
Bijective function: A bijective function is a type of function that establishes a one-to-one correspondence between elements of two sets, meaning that every element in the first set is paired with exactly one unique element in the second set and vice versa. This property ensures that both the function is injective (no two inputs map to the same output) and surjective (every possible output is covered). Bijective functions are crucial in understanding concepts like inverses and equivalence between sets.
Bipartite graph: A bipartite graph is a type of graph where the set of vertices can be divided into two distinct groups such that no two graph vertices within the same group are adjacent. This characteristic makes bipartite graphs useful for modeling relationships in various scenarios, such as matching problems and network flow. The two groups are often referred to as partite sets, and edges only connect vertices from different sets.
Chemical Compound: A chemical compound is a substance formed when two or more different elements bond together in a fixed ratio, resulting in unique properties that differ from those of its constituent elements. These compounds can be classified into various categories, such as ionic and covalent compounds, depending on the nature of the bonds formed between the atoms.
Complement: The complement of a set A, denoted by A', consists of all elements not in A but within the universal set U. The universal set U contains all possible elements under consideration.
Complement: In set theory, the complement of a set refers to all the elements in the universal set that are not included in that specific set. Understanding the complement is crucial as it helps in visualizing and analyzing relationships between sets, especially when using diagrams, performing operations with two or three sets, applying De Morgan's laws, and calculating probabilities.
Complete Graph: A complete graph is a type of graph in which every pair of distinct vertices is connected by a unique edge. This structure ensures that there are no missing connections, resulting in a highly interconnected network. The concept of complete graphs is crucial for understanding graph theory, as they serve as a benchmark for comparing the connectivity and efficiency of other graph structures.
Cycle: A cycle in graph theory is a path that starts and ends at the same vertex with no other vertices repeated. It forms a closed loop or circuit within the graph structure.
Cycle: A cycle in graph theory refers to a path that starts and ends at the same vertex, containing at least one edge, with no other repetitions of vertices and edges. This concept is crucial for understanding how graphs are structured and navigated, as cycles can affect various properties and operations within a graph, including traversal and connectivity.
Cyclic subgraph: A cyclic subgraph is a type of subgraph that contains a cycle, meaning there is a closed path in the graph where no edges or vertices are repeated except for the starting and ending vertex. This concept plays a significant role in understanding the structure of larger graphs, as it helps identify relationships and connectivity among vertices. Cyclic subgraphs are important for analyzing properties such as graph traversal and connectivity in various contexts.
Degree: In mathematics, a degree is a unit of measurement for angles, denoting the size of the angle in a circle. It also represents the number of edges connected to a vertex in graph theory, providing insight into the structure and navigation of graphs. Understanding degrees helps clarify relationships between angles and graph structures, which is essential for analyzing various mathematical concepts.
Degree sequence: A degree sequence is a list of vertex degrees of a graph, usually arranged in non-increasing order. It provides valuable insights into the structure and properties of a graph, such as connectivity and the possibility of certain types of graphs existing with that sequence. Understanding degree sequences helps in comparing different graphs and analyzing their characteristics.
Edge: An edge is a connection between two vertices in a graph. It can be directed (with a direction) or undirected (without a direction).
Edge: An edge is a fundamental component of a graph that represents a connection or relationship between two vertices (or nodes). Edges can be directed or undirected, and they can have weights associated with them, representing the cost or distance between the connected vertices. Understanding edges is crucial for analyzing graph structures, navigating graphs, and exploring various properties like circuits and paths.
Graph Automorphism: A graph automorphism is a mapping of a graph onto itself that preserves the structure of the graph, meaning it keeps the vertices connected in the same way. This concept is essential for understanding the symmetries within graphs, as automorphisms reveal how the graph can be transformed without altering its essential properties. Graph automorphisms are crucial for comparing graphs and analyzing their structural features.
Graph connectivity: Graph connectivity refers to the minimum number of elements (like vertices or edges) that need to be removed to disconnect the remaining vertices from one another. This concept is crucial because it helps to determine how resilient a graph is to disruptions and how information flows through it. The degree of connectivity can reveal a lot about the overall structure of the graph, influencing aspects like the existence of Euler circuits and making it essential when comparing different graphs.
Graph invariant: A graph invariant is a property of a graph that remains unchanged under specific transformations or operations, such as isomorphisms. This concept is crucial for comparing different graphs, as it allows us to identify when two graphs are structurally the same despite being drawn differently. Understanding graph invariants helps in analyzing the characteristics and relationships of graphs in various mathematical and practical applications.
Graph Theory: Graph theory is a branch of mathematics that studies the properties and relationships of graphs, which are mathematical structures used to model pairwise relationships between objects. It provides essential tools for analyzing various structures and processes, from social networks to transportation systems. Understanding graph theory helps in exploring complex connections, paths, and cycles within different contexts.
Isomorphic: Isomorphic refers to two graphs that can be transformed into each other by renaming vertices without changing the structure or connections. Essentially, they have an identical form or shape in terms of vertex connections and edges.
Isomorphic: Isomorphic refers to a relationship between two structures, such as graphs, where there exists a one-to-one correspondence between their elements that preserves the relationships among those elements. In the context of comparing graphs, this means that two graphs are isomorphic if there is a way to redraw one graph to look exactly like the other while keeping the connections intact, showcasing their structural similarity despite potential differences in representation.
Isomorphism: Isomorphism is a concept in mathematics that describes a structural similarity between two objects, such that there exists a one-to-one correspondence between their elements while preserving relationships. This idea is crucial when comparing graphs, as it helps identify whether two graphs can be considered the same in terms of their structure, despite differences in their appearance or labeling. Essentially, if two graphs are isomorphic, they can be transformed into one another through relabeling of vertices without changing the connectivity of the graphs.
Nonplanar: A graph is nonplanar if it cannot be drawn on a plane without edges crossing. Nonplanar graphs cannot be embedded in the plane without edge intersections.
One-to-one correspondence: One-to-one correspondence is a relationship between two sets where each element of the first set is paired with exactly one unique element of the second set, and vice versa. This concept is fundamental in understanding the idea of equivalence between different sets, particularly when comparing their sizes or cardinalities. It highlights how two collections can be matched perfectly without any overlaps or omissions, providing a clear way to analyze and compare their structures.
Ordered Pair: An ordered pair is a pair of elements where the order of the elements matters, typically represented as (x, y) in a coordinate system. The first element, x, is known as the x-coordinate, and the second element, y, is known as the y-coordinate. This concept is crucial for representing points in a two-dimensional space, which directly connects to solving equations and inequalities involving two variables.
Planar: A planar graph is a graph that can be drawn on a plane without any edges crossing. It must be possible to rearrange the vertices and edges so that no two edges intersect except at their endpoints.
Social network: A social network is a structure made up of individuals or organizations, known as 'nodes', that are connected by one or more types of relationships, such as friendship, kinship, or professional ties. These networks can be represented graphically, allowing for the analysis of the connections and interactions between members, revealing important patterns in social behavior and communication.
Square: A square is a special type of polygon that has four equal sides and four right angles, making it a specific case of a rectangle and a rhombus. This geometric shape is significant in various contexts, as it serves as a fundamental building block for calculating area and comparing graphical data. Its properties allow for clear calculations and comparisons when dealing with two-dimensional space and data visualization.
Square meter: A square meter (m²) is the SI unit of area, defined as the area of a square with sides that are one meter in length. It is commonly used to measure surface areas in various applications such as land, rooms, and other spaces.
Star graph: A star graph is a type of graph structure where one central node is connected directly to all other nodes, resembling a star shape. This configuration makes it an important model for understanding relationships and connections in various systems, as it highlights the centrality of a single point while maintaining simplicity in its layout.
Triangle: A triangle is a polygon with three edges and three vertices, and it is one of the simplest shapes in geometry. The sum of the internal angles of a triangle always equals 180 degrees, which is a fundamental property that connects it to various mathematical concepts like perimeter, area, and graphing relationships.
Vertex: A vertex is a point where two or more curves, lines, or edges meet. In different contexts, it can represent a significant feature such as the peak of a parabola, a corner of a polygon, or a key point in graph theory. Understanding the concept of a vertex helps in analyzing the properties and relationships of various mathematical structures.
Vertex set: The vertex set of a graph is the complete collection of all its vertices, which are the fundamental units that represent points in a graph. This set is crucial as it provides the foundational elements needed to analyze and understand the structure of the graph, including its connectivity, paths, and overall characteristics.
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