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Walks and graphs are fundamental concepts in graph theory, which is a branch of mathematics that studies the properties and applications of graphs. A walk is a sequence of vertices connected by edges, while a graph is a collection of vertices and edges that represent a network of relationships.
Walks and graphs have numerous applications in various fields, including computer science, biology, social sciences, and transportation planning. For example, they can be used to model social networks, analyze transportation networks, and optimize computer algorithms.
A graph can be represented using an adjacency matrix or an adjacency list. An adjacency matrix is a two-dimensional array where each element represents an edge between two vertices. An adjacency list is a collection of linked lists where each list contains the vertices that share an edge.
A walk can visit the same vertex or edge multiple times, while a path cannot. In other words, a path is a type of walk that does not contain any repeated vertices or edges. Additionally, a walk can be open (ends at a different vertex than it starts) or closed (ends at the same vertex it starts).
The shortest path problem is a fundamental graph theory problem that aims to find the most efficient path between two vertices in a graph. This problem is commonly solved using algorithms like Dijkstra's algorithm or the A* search algorithm. It has many applications, such as finding the shortest route between two locations on a map or the most efficient way to travel between cities.
