The post Uniqueness of Minimum Spanning Tree appeared first on Educative Site.

]]>*Fig: Minimum Spanning Tree*

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]]>The post Graph Theory and Applications appeared first on Educative Site.

]]>Graph theory is the study of graphs which are mathematical structures used to model pairwise relations between objects.

- Development of graph algorithm.
- Chemical Identification.
- Combinatoric operations research.
- Graph coloring.
- Job scheduling.
- Aircraft scheduling.
- Bi-processor tasks.
- Time-table scheduling.
- Map coloring.
- GSM mobile phone networks.
- Computer Network security.
- Ad-hoc networks.
- Symbol recognition.
- Modeling sensor network.
- Pattern recognition.

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]]>The post Euler Tour – Euler Trail – Hamiltonian Cycle all graph appeared first on Educative Site.

]]>Conditions:

- At most 2 odd degree (number of odd degree <=2) of vertices.
- Start and end nodes are different.

Conditions:

- All vertices have even degree.
- Start and end node are same.

Conditions:

- All edges are traversed exactly once.
- Some nodes (vertices) are traversed more than once.

Conditions:

- All nodes are traversed exactly once.
- Some edges are traversed more than once.

Conditions:

- Vertices have at most two odd degree.
- Some nodes are traversed more than once.
- Start and end node is not same.

Conditions:

- Start and end node is same.
- Some edges is not traversed or no vertex has odd degree.

**This graphs are very very important for any examinations.**

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]]>The post Euler Tour – Euler Trail appeared first on Educative Site.

]]>In a graph, cycle is a tour with start and end with same node.

Trail is a path where every edge (line) is traversed exactly once and start and end vertices (node) are different.

Here 1->3->8->6->3->2 is trail and also 1->3->8->6->3->2->1 will be a closed trail.

Euler tour is a graph cycle when every edge is traversed exactly once but nodes (vertices) may be visited more than once and all vertices have even degree with start and end node is the same.

*Fig: Euler Tour*

Euler trail is a graph path when every edge is traversed exactly once but nodes (vertices) may be visited more than once and at most 2 vertices have odd degree with start and end node is the different.

*Fig: Euler Trail*

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]]>The post Hamilton Path and Hamilton Cycle appeared first on Educative Site.

]]>A Hamilton path is a path in a graph that visits every node (vertices) exactly once in a way that start and end node is different.

A Hamilton cycle in a graph is a cycle that visits every node (vertices) exactly once in a way that start and end node is same.

Fig: The cycle a – b – e – c – d – f – g is Hamiltonian cyle |

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]]>The post Star Graph – Face – Planar Embedding appeared first on Educative Site.

]]>Star graph is a complete bipartite graph K, a tree with internal node and K leaves.

*Fig: Star Graph*

In a planar drawing of a graph, the edges divide up the plane into connected regions. These regions are called faces.

A planar embedding of a connected graph consists of a non-empty set of closed walks of the graph called the discrete faces of the embedding.

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]]>The post Tree – Tree Properties and Minimum Spanning tree appeared first on Educative Site.

]]>A connected acyclic graph is called a tree.

- Any connected sub-graph is a tree.
- There is a unique simple path between every pair of vertices.
- Adding an edge between non-adjacent nodes in a tree creates a graph with a cycle.
- Removing an edge disconnects the graph.
- If the tree has at least two vertices, then it has at least two leaves.
- The number of vertices in a tree is one larger than the number of edges.

Spanning tree is a tree in a connected graph that contains a sub-graph with the same vertices as the graph. This is called a spanning tree for the graph. To find the spanning tree with minimum weight is called the minimum weight spanning tree (MST).

*Fig: MST in a Graph*

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]]>The post Chromatic Number and applications appeared first on Educative Site.

]]>Chromatic Number of a graph is the smallest number of colors needed to color the vertices so that no two adjacent vertices share the same color.

- Scheduling tasks
- Register allocations
- Server updating
- Pattern matching
- Solving suduko (9 coloring)
- Bipartite graph
- Map coloring etc.

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]]>The post Walk – Cycle – Connectivity – Tournament Graph appeared first on Educative Site.

]]>A walk is a sequence of vertices.

Cycle is a closed walk in a graph, a closed sequence of vertices.

A graph is said to be connected when every pair of vertices are connected.

A graph that represents the result of round robin tournament.

*Fig: Tournament Graph*

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]]>The post Diagraph – Directed Walk – Directed acyclic graph (DAG) appeared first on Educative Site.

]]>A graph with directed edges is called a directed graph or diagraph.

*Fig: A 3 node directed graph*

A directed walk (or Walk) in a directed graph is a sequence of vertices v0, v1, v2………vk and egdes v0 -> v1, v1 -> v2, ……. vk-1 -> vk.

A directed graph is called a Directed Acyclic Graph (DAG) if it does not contain any directed cycles.

*Fig: DAG*

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