Loading...

The Regular Graph Retated to Rings

Jamaali, Mohsen | 2011

633 Viewed
  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 42669 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Science
  6. Advisor(s): Akbari, Saeed
  7. Abstract:
  8. There are many papers in which some graphs are assigned to algebraic structures such as rings groupsThe concept of regular graph related to a ring was rst investigated by DF Anderson and A Badawi in Assume that R is a commutative ring and Z??R denotes the set of zerodivisors of R and Reg??R R n Z??R The regular graph of R which is denoted by Reg????R is a graph whose vertex set is Reg??R and two vertices x and y are adjacent if and only if x y ?? Z??R This can be generalized to a non commutative ring For the vertex set we consider the set of left ??right zerodivisors and join two elements if their sum is a left ??right zerodivisor Let R be the ring of n n matrices over a eld F with charactristics not equal to It is shown that the clique number of R is nite Furthermore for n and an arbitrary eld the exact value of the clique number is determined Also some theorems about clique number of some rings are given Another question which arises naturally is the determi nation of the chromatic number of regular graphs The latter question was presented at BCC Since nding the exact value of the chro matic number seems to be dicult we concentrate on simpler cases For example when G is a soluble subgroup of GLn??F it is shown that the chromatic number of the subgraph ??Abstract of regular graph induced on the elements of G is nite Also some exact bounds for the clique number are obtained for a cyclic subgroup of GLn??F The set of the neighbors of a vertex is also investigated Let N??A denote the set of nonsingular matrices whose sum with A is singular and N??B be dened similarly For example it is shown that for an arbitrary eld F if AB ?? GLn??F and N??A N??B then A B Then the concept of the inner products arises naturally Let V be an inner product vector space We associate the inner product graph G??V to it as follows The vertices consist of the elements of V and two vertices u and v are adjacent if and only if uv Some theorems about the clique number and chromatic number of G??V are proved As an example for V Zp Zp ??p the clique number is determined exactly
  9. Keywords:
  10. Regular Graph ; Matrix Algebra ; Clique Number ; Chromatic Number ; Determinants

 Digital Object List

 Bookmark

No TOC