Sharif Digital Repository

Let (P;≼) be a partially ordered set (poset, briefly) with a least element 0. In this thesis, we deal with zero-divisor graphs of posets. We show that if the chromatic number r(P) and the clique number r(P) (x(r(P)) and !(r(P)), respectively) are finite, then x(r(P)) = !(w(P)) = n in which n is the number of minimal prime ideals of P. We also prove that the diameter of such a graph is either 1, 2 or 3 while its girth is either 3, 4 or 1  

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... 

The aim of this thesis is to introduce some algebraic topologies methods and apply them on fining the chromatic number of some famous graphs and also hypergraphs. In the first part, we will use a mixture of two well-known technics, Tucker lemma and Discrete Morse theory to find an upper bound for the chromatic number of s-stable Kneser for some specific vector s. to find the sharper upper bound, we will deviate our strategy and use another approach by finding an edge-labeling and apply some theorems in POSET algebraic topology. In this way, we also find a connection between Young diagrams and the numbers of spheres in the box complex related to Kneser graphs and hypergraph. Actually, we can... 

In this thesis, we study some connections between the graph-theoretic and algebraic properties of co-maximal graph of algebraic structures. We follow two purposes. First, what properties of algebraic structures can be found from co-maximal graph of algebraic structures. Second, what geometric or graph theoretical properties of co-maximal graph of algebraic structures can be found from specefic algebraic structures. Let G be a group and I(G)∗be the set of all non-trivial sub-groups of G. The co-maximal graph of subgroups of G, denoted byΓ(G), is a graph with the vertex set I(G)∗and two distinct vertices H and K are adjacent if and only if HK=G. We char-acterize all groups whose co-maximal...