Full Record

Author | Changiz Rezaei, Seyed Saeed |

Title | Entropy and Graphs |

URL | http://hdl.handle.net/10012/8173 |

Publication Date | 2014 |

University/Publisher | University of Waterloo |

Abstract | The entropy of a graph is a functional depending both on the graph itself and on a probability distribution on its vertex set. This graph functional originated from the problem of source coding in information theory and was introduced by J. K\"{o}rner in 1973. Although the notion of graph entropy has its roots in information theory, it was proved to be closely related to some classical and frequently studied graph theoretic concepts. For example, it provides an equivalent definition for a graph to be perfect and it can also be applied to obtain lower bounds in graph covering problems. In this thesis, we review and investigate three equivalent definitions of graph entropy and its basic properties. Minimum entropy colouring of a graph was proposed by N. Alon in 1996. We study minimum entropy colouring and its relation to graph entropy. We also discuss the relationship between the entropy and the fractional chromatic number of a graph which was already established in the literature. A graph $G$ is called \emph{symmetric with respect to a functional $F_G(P)$} defined on the set of all the probability distributions on its vertex set if the distribution $P^*$ maximizing $F_G(P)$ is uniform on $V(G)$. Using the combinatorial definition of the entropy of a graph in terms of its vertex packing polytope and the relationship between the graph entropy and fractional chromatic number, we prove that vertex transitive graphs are symmetric with respect to graph entropy. Furthermore, we show that a bipartite graph is symmetric with respect to graph entropy if and only if it has a perfect matching. As a generalization of this result, we characterize some classes of symmetric perfect graphs with respect to graph entropy. Finally, we prove that the line graph of every bridgeless cubic graph is symmetric with respect to graph entropy. |

Subjects/Keywords | Entropy; Probability distribution; Graphs; Fractional Chromatic Number; Vertex packing polytope; Perfect Graphs; Line Graphs |

Language | en |

Country of Publication | ca |

Record ID | handle:10012/8173 |

Repository | waterloo |

Date Retrieved | 2019-06-25 |

Date Indexed | 2019-06-26 |

Sample Search Hits | Sample Images

…21
3.5
Entropy of Some Special Graphs . . . . . . . . . . . . . . . . . . . . . . . .
23
3.6
Graph Entropy and Fractional *Chromatic* *Number* . . . . . . . . . . . . . .
28
3.7
Probability Density Generators…

…entropy and perfect graphs and fractional
*chromatic* *number* of a graph. Chapter 4 is devoted to minimum entropy colouring of
a given graph and its connection to the graph entropy. G. Simonyi in [36] showed that
the maximum of the graph entropy of…

…a given graph over the probability density of its
vertex set is equal to its fractional *chromatic* *number*. We call a graph is symmetric with
respect to graph entropy if the uniform density maximizes its entropy. We show that vertex
transitive graphs…

…with its fractional *chromatic* *number* in more detail. Furthermore, we obtain the
fractional *chromatic* *number* of a vertex transitive graph using its properties along with
the Lagrange Multiplier technique in convex optimization.
3.1
Entropy of a Convex…

…F ) we denote by F [Z] the induced subgraph of F on Z. The
*chromatic* *number* of F is denoted by Ï‡(F ).
Let
T (n) = {U âŠ† V n : P n (U ) â‰¥ 1 âˆ’ }.
We define the functional H(G, P ) with…

…33
3.8
Additivity and Sub-additivity . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.9
Perfect Graphs and Graph Entropy . . . . . . . . . . . . . . . . . . . . . .
36
ix
4 *Chromatic* Entropy
39
4.1
Minimum Entropy Coloring…

…39
4.2
Entropy Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.3
*Number* of Colours and Brooksâ€™ Theorem . . . . . . . . . . . . . . . . . . .
43
4.4
Grundy Colouring and Minimum Entropy Colouring…

…information theoretic functional which is defined on a graph
with a probability density on its vertex set. This functional was originally proposed by J.
K¨
orner in 1973 to study the minimum *number* of codewords required for representing an
information source…