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okay, let's dive into a comprehensive tutorial on important topics from graph theory and queuing theory (gtq) relevant to the gate 2025 algorithms syllabus, with a focus on problem-solving approaches, cs made easy methods, and code examples (where applicable).

*i. graph theory (gt)*

graph theory is a fundamental branch of mathematics and computer science that deals with the study of graphs. a graph is a structure consisting of a set of vertices (nodes) and a set of edges that connect these vertices. graphs are used to model relationships between objects, and they are widely used in computer science, mathematics, engineering, and other fields.

*1. basic definitions and representation:*

*graph:* g = (v, e), where v is the set of vertices (nodes) and e is the set of edges.
*vertices (nodes):* the fundamental units of a graph.
*edges:* connect pairs of vertices. edges can be directed (one-way) or undirected (two-way).
*directed graph (digraph):* edges have a direction. e consists of ordered pairs of vertices (u, v), where the edge goes from u to v.
*undirected graph:* edges do not have a direction. e consists of unordered pairs of vertices {u, v}.
*weighted graph:* edges have associated weights (costs, distances, capacities).
*adjacency:* two vertices are adjacent if they are connected by an edge.
*degree (undirected):* the number of edges incident to a vertex.
*in-degree (directed):* the number of edges entering a vertex.
*out-degree (directed):* the number of edges leaving a vertex.
*path:* a sequence of vertices connected by edges.
*cycle:* a path that starts and ends at the same vertex.
*connected graph:* there is a path between every pair of vertices.
*complete graph:* every pair of vertices is connected by an edge.
*tree:* a connected graph with no cycles.
*spanning tree:* a subgraph of a connected graph that is a tree and includes all vertices ...

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