Calculating Nicole's Profit: Accounting vs. Economic Profit Analysis, Schemes and Mind Maps of Mathematics

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Discrete Mathematics

Graph

Hà Minh Hoàng

Faculty of Data Science and Artificial Intelligence

College of Technology

National Economics University

● (^) Graphs and Their Classifications ● (^) Graph Terminologies ● (^) Graph Representations and Isomorphism ● (^) Paths and Connectivity ● Eulerian and Hamiltonian Paths ● (^) Shortest Path Problem ● (^) Planar Graphs ● (^) Graph Coloring Content

• (^) A simple graph 𝐺=( 𝑉 𝐸, ) consists of a non-empty set 𝑉whose elements are called vertices, and a set 𝐸whose elements are called edges. Each edge is an unordered pair of distinct vertices. Classification: simple graph

• (^) A multigraph 𝐺=( 𝑉 𝐸, ), where 𝑉 is the set of vertices and 𝐸is the set of edges, is an undirected graph that may have multiple edges connecting the same pair of vertices (called parallel edges or multiple edges).
• (^) A simple graph is a special case of a multigraph. Classification: multigraph

• (^) A pseudograph 𝐺=( 𝑉 𝐸, ), where 𝑉 is the set of vertices and 𝐸is the set of edges, is a graph that allows edges connecting a vertex to itself (called loops).
• (^) A pseudograph can have multiple edges and loops. Classification: pseudograph

• (^) A simple directed graph 𝐺=( 𝑉, A ), where 𝑉is a non-empty set whose elements are called vertices, and A is a set whose elements are called arcs, consists of ordered pairs of distinct vertices. Classification: simple directed graph

Graph terminology

Example 1. Niche overlap graph in ecology.

• Graphs are used in many models that consider the interactions between species, e.g., the competition among species in an ecosystem can be modeled using a "niche" graph.
• Each species is represented by a vertex. An undirected edge connects two vertices if the corresponding species compete with each other (i.e., they share the same food source).
• From this graph, squirrels and raccoons compete with each other, while crows and shrews do not. Examples: Biological networks

Example 2. Protein interaction graph.

• (^) The protein interaction graph of a cell is extremely large and complex.
  • (^) Yeast cells have more than 6,000 proteins, and more than 80,000 interactions
  • (^) Human cells have more than 100,000 proteins, with perhaps 1,000,000 interactions between them.
• Additional vertices and edges are added to a protein interaction graph when new proteins and interactions between proteins are discovered.
• The protein interaction graphs are often split into smaller graphs called modules that represent groups of proteins that are involved in a particular function of a cell.
• Figure illustrates a module of the protein interaction graph, comprising the complex of proteins that degrade RNA in human cells. Examples: Biological networks

Example 2. Influence Graph

• When studying the personalities of a group of people, it is observed that some individuals may influence the thoughts of others.
• A directed graph, known as an influence graph, can be used to model this problem.
• Each person in the group is represented by a vertex.
• If a person represented by vertex 𝑎can influence a person represented by vertex 𝑏 , then there is a directed arc from vertex 𝑎 to vertex 𝑏. Examples: Social networks

Call graphs: Examples: Communication networks

Example 3. Citation graph

• A citation graph is a directed graph where each node represents a scientific paper, and each directed edge points from one paper to another that it cites.
• These graphs form the backbone of scholarly communication and offer powerful insights into the structure and evolution of scientific knowledge.
• The Web graph Examples: Information graphs

• (^) Airline routes
• (^) Road networks
• (^) Rail networks
• (^) Shipping networks Examples: Transportation networks

• (^) Module dependency graphs
• (^) Precedence graph and concurrent processing Examples: Software design applications