Graph Theory: Understanding Graphs, Minimum Spanning Trees, and Graph Coloring, Exercises of Discrete Mathematics

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Modelling using

network graph

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What is a Graph?

 (^) Graph G = (V,E) ( simple, Undirected)

 (^) Set V of vertices ( nodes ): A non-empty finite set

 (^) Set E of edges: E  V (^2) , 2-element subsets of V.

 (^) Elements of E are unordered pairs {v,w} where v,w 

V.

 (^) So a graph is an ordered pair of two sets V and E, such

that E  V^2.

 (^) Variations: Directed graph, ( E  V  V)

 (^) Weighted graph

 (^) W: ER, c: V R

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Undirected

Graphs — Weighted graphs

Vertices (aka nodes)

Edges

Weights

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Directed Graph (digraph)

Vertices (aka nodes)

Edges

Input: A weighted connnected graph.

Idea: Maintain an acyclic subgraph H, enlarging it by edges with low

weight to form a spanning tree. Consider edges in non-decreasing order of

weight, breaking ties arbitrarily.

Initialization: Set E(H) = ∅

Iteration: If the next cheapest edge joins two components of H, then

include it; otherwise, discard it. Terminate when H is connected.

Output : A minimum spanning tree H of G.

Kruskal’s Algorithm to find MST

INITIAL ITERATION

FINAL ITERATION

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MST Applications

Electrical

wiring

of a house

using

minimum

amount of

wires

(cables)

power outlet

or light

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MST Applications

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Graph

Representation

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Minimum

Spanning

Tree

for

electrical

eiring

Each subject is marked as a node and there

is a edge between all those subjects(nodes)

which clash with each other.

The coloring of the graph obtained above

give us time-tabeling as each color

represents a time slot.

Application of graph coloring in lecture scheduling

Dijkstra’s shortest path algorithm

Given a graph and a source vertex in the graph, find shortest

paths from source to all vertices in the given graph.This problem

is called shortest path problem

Assumption: All edge weights are non-negative.

Working:

Maintain a set S of vertices whose final shortest-path weights from

the source s have already been determined.

Repeatedly select the vertex u ϵ V−S with the minimum

shortest-path estimate.

Add u to S, and relaxes all edges leaving u.

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Computer Network



Problem :

 How to find a path from the source router to

destination router so that the message takes

least time?

Solution -Shortest path problem

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Automation of mechanical drilling

 (^) A drilling head drills several, say n, holes in a plate ( Spare

parts of automobiles0.

 The drilling head moves from one point to another and drills

holes.

 (^) How to minimize the head movement time so that

productivity increases?

 (^) Model: G=(V,E), V= the n holes. E= ij, for every hole i to

every hole j. W(ij)=actual distance travel by the drilling head.

 Solution: TSP- find an ordering of vertices, a tour, having

minimum cost.