Loading...

A Study on the Shannon Zero Error Capacity of Graphs

Izadi Moein, Ali | 2023

81 Viewed
  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 56665 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Jafari, Amir; Alipour, Sharareh
  7. Abstract:
  8. The issue of reliability, which is associated with a low probability of error in a communication system, is an important parameter in information theory. By reliability, we mean a low decoding error probability, and one of the main criteria for this is the zero decoding error probability. In the zero-error probability, the decoding error probability for any code is exactly zero. Error-free capacity was examined by Claude Shannon in 1956. The Shannon capacity of a communication channel is defined using the independent sets of strong products of graph. Shannon calculated the capacity of graphs up to five vertices except the graphC5. Lovász, in 1979, using the upper bound he provided, was able to calculate the Shannon capacity of the graphC5.After the problem was posed by Shannon, various approaches were taken to solve it, one of which was to find upper and lower bounds for Shannon capacity. In general, finding the Shannon capacity is very difficult, and there are many graphs for which we do not know how to find their Shannon capacity. The most important example of these graphs are odd cycles with a length of at least seven, especially finding the capacity of the graphC7is one of the famous open problems in combinatorics. In this thesis, we study the concept of Shannon capacity of graphs. We investigate the existing upper bounds for Shannon capacity, including fractional independence number, Lovász’s number, Haemers’s bound and fractional Haemers’s number. we also present examples of universal graphs that are not perfect. Finally, we examine the existing lower bounds for Shannon capacity, especially regarding odd cycle graphs
  9. Keywords:
  10. Independence Number ; Shannon Capacity ; Lovász’s Number ; Perfect Graph ; Universal Graph ; Fractional Independence Number

 Digital Object List

 Bookmark

  • مقدمه
  • ظرفیت شانون
    • تعاریف مورد نیاز
    • ظرفیت شانون
    • نظریه طیفی گراف‌ها
    • طیف گراف‌ها
    • کران طیفی برای عدد استقلال
    • طیف ضرب قوی گراف‌ها
  • گراف‌های عام
    • گراف‌های عام
    • گراف‌های تام
    • مثال‌هایی از گراف‌های عام که تام نیستند
  • کران‌های بالا برای ظرفیت شانون
    • عدد لووس
    • عدد لووس گراف C5
    • عدد لووس دورهای فرد
    • عدد لووس گراف‌های کنسر
    • کران همرز
    • کران همرز کسری
  • کران‌های پایین برای ظرفیت شانون
    • ضرب دورهای فرد
    • کران پایین ظرفیت شانون برای گراف C7
    • کران پایین برای گراف C9
    • کران پایین برای گراف‌های دور فرد
    • عدد استقلال و کران پایین ظرفیت شانون برای دسته‌ی خاصی از گراف‌های دوری
  • پیشنهادها برای ادامه کار
  • مراجع
  • واژه‌نامه
...see more