Some Concepts of Graph TheoryMalay BhattacharyyaSRF, MIU, ISI Kolkata

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The story begins…

p königsberg bridge on the Pregel River

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The königsberg bridge problem

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Find a traversal through the cities thatwould cross each bridge once andonly once.Formalized version:Find an Euler walk in the shown graph.

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A graph

p G = (V, E)n V: Set of vertices {v1, v2, …, vm}n E: Set of edges {e1, e2, …, en}

n E ⊆ V⨯V

p Order of the graph: mp Size of the graph: n

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Vertex incidence, self-loops and degree

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Given, G = (V, E)I(v) = {(i, j)⊆V⨯v|(i, j)ϵE}d(v) = |I(v)|d(G) = (1/|V|).∑vϵV d(v)∑vϵV d(v) is twice the size of the graph i.e. 2|E|

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Directed graph

p Distinguishing vertex pairs

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In-degree and out-degree

p In-degree: d–(v)p Out-degree: d+(v)

∑vϵV d–(v) = ∑vϵV d+(v) = |E|

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Significant theorems

p The number of vertices of odd degree in a graph is always even.

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Walks, paths and cyclesp A walk is a sequence of edges in a graph

n A walk in which no edge is repeated is a trailn A trail in which no vertex is repeated is a pathn A path which have same start and end vertices is a cycle

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Tree

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Properties of a tree

p A tree is a connected graph with no cyclesp There is one and only one path between every pair of verticesp The degree of a tree with n vertices is 2(n–1)/n

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Cliques: Complete subgraphs

1-vertex cliques (vertices) – 23,2-vertex cliques (edges) – 42,3-vertex cliques (light blue triangles) – 19 (11 are maximal),4-vertex cliques (dark blue trianges) – 2 (both are maximum and maximal).Clique number of the graph: 4

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Bipartite graph and biclique

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G = (V1, V2, E), where E ⊆ V1⨯ V2

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Planar and non-planar graphs

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The story ends…

p An Euler walk is a walk that traverses every edges of a graph oncen Criterion: During any walk in the graph, the number of times one enters a non-terminal vertex equals the number of times one leaves it.n Thus, the degree values of thenon-terminal vertices should beeven.n But, this graph has degreevalues 3, 3, 5, 5.n So, no solution exists.

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References1. F. Harary, Graph Theory, Addison-Wesley, 1969.2. N. Deo. Graph Theory with Application to Engineering and Computer

Science, Prentice-Hall, Englewood Cliffs, N.J., 1974.3. C. Berge, Graphs, North-Holland, 1985.4. R. J. Trudeau, Introduction to Graph Theory, Dover Publications, 1994.5. D. B. West, Introduction to Graph Theory, Prentice Hall, 1996.6. Introduction to Graph Theory by R.J. Wilson Addison WesleyLongman 1996.7. R. Diestel. Graph Theory, Third Edition, Springer, Heidelberg, 2000.8. M. C. Golumbic, Algorithmic Graph Theory and its Applications, SecondEdition, Annals of Discrete Mathematics, 57, 2004.May 12, 2011 16Interactive Seminar Series 2011

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Thank you

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