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Cayley Graphs and Annihilating-Ideal Graph of a Ring

Aalipour Hafshejani, Ghodratollah | 2013

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 45019 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Akbari, Saeed
  7. Abstract:
  8. In the recent years, the role of combinatorics and graph theory have grown in the progress of computer sciences. For instance, the circulant graphs have applications in design of interconnection networks and the graphs with integer eigenvalues are applied in modelling quantum spin networks supporting the perfect state transfer. The circulant graphs with integer eigenvalues also found applications in molecular graph energy. In 2006, it was shown that an n-vertex circulant graph G has integer eigenvalues if G=Cay(Zn; T ) or G= Cay(Zn; T)∪Cay(Zn;U(Zn)), where T Z(Zn). The Cayley graph Cay(Zn;U(Zn)) is known as the unitary Cayley graph. Fuchs defined the unitary Cayely graph of a commutative ring R, Cay(R+;U(R)), as a generalization of the unitary Cayley graph. By generalizing the definition of Xn(D n f1g)=Cay(Zn;Z(Zn)) to a commutative ring R, we study Xn(D n f1g). This generalization can be simply done by CAY(R) = Cay(R+;Z(R)), a graph whose vertices are elements of R and two distinct vertices x and y are joined by an edge if and only if x y 2 Z(R). Recently, the energy of Cay(Zn;Z(Zn)) and its other properties were studied. In this thesis, we consider Cay(R;Z(R)) for an arbitrary commutative ring R. Studying this graph is important since the algebraic properties of the ring R can be investigated through combinatorial properties of Cay(R;Z(R)). This approach is recently applied for studying other algebraic structures. For instance, the annihilating-ideal graph of a commutative ring R is another graph associated with R. It was conjectured that if jMin(R)j 3, then girth(AG(R)) 3. Here, we give a stronger result by proving that for every commutative ring R, !(AG(R)) jMin(R)j
  9. Keywords:
  10. Prime Ideal ; Chromatic Number ; Clique Number ; Commutative Ring ; Zero-Divisor Graph ; Eigen Values ; Topology

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