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In this thesis, we study some bounds for the vertex chromatic num- ber and edge chromatic number of a graph. One of the most fa- mous theorems on graph colorings is Brooks’ Theorem, which asserts that every connected graph with maximum degree ∆(G) is ∆(G)- colorable unless G is an odd cycle or a complete graph. The following result has been proved: If every vertex of a graph G lies on at most k odd cycles for some nonnegative integer k, then χ(G) 1+√8k+9 . We recall from Vizing’s Theorem that the edge chromatic number of any graph must be equal either to ∆(G) or ∆(G) + 1. In this thesis, families of graphs that are Class 1 or Class 2 will be introduced.
 

In 1975 a Chemist Milan Randić proposed a concept named Randić index which is defined as follows: This index is generalized by replacing any real number α with which is called the general Randić index. Let G be a graph of order n. Erdős and Bollobás showed the lower bound for Randić index, Also, an upper bound for Randić index is n/2. In 2018 Suil O and Yongtang Shi proved a lower bound with minimum and maximum degree of a graph. They have shown for graph G we have, R(G) Also, a relation between Randić index and the energy of the graph has found. Indeed, it was proved that E(G) ⩾ 2R(G), where E(G) is the energy of graph. Many important bounds related to graph parameters for Randić index... 

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... 

A Latin square of order n, is an n n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column.Two cells of a Latin square are independent when they are not in the same row, or in the same column and they are not the same symbol. If they are not independent, they are called dependent. A k-coloring of a Latin square is assigning k colors to its cells where no two dependent cells have the same color. The smallest k for which we have a k-coloring for a Latin square L with k colors is called chromatic number of L and we denote it by L). If we consider the Cayley table of an arbitrary finite group of order n, then we have a Latin square of... 

We study Beck-like coloring of partially ordered sets (posets) with a least element 0. To any poset P with 0 we assign a graph (called a zero-divisor graph) whose vertices are labelled by the elements of P with two vertices x, y adjacent if 0 is the only element lying below x and y. We prove that for such graphs, the chromatic number and the clique number coincide.Also, we give a condition under which posets are not finitely colorable  

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... 

We study colorings of quasiordered sets (qosets) with a least element 0. To any qoset Q with 0 we assign a graph (called a zerodivisor graph) whose vertices are labelled by the elements of Q with two vertices x; y adjacent if the only elements lying below x and y are those lying below 0. We prove that for such graphs, the chromatic number and the clique number coincide  

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 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... 

One of the interesting and active area in the last decade is using graph theoretical tools to study the algebraic structures. In this thesis, first we study the intersection graphs of non-trivial submodules of a module, their clique number and their chromatic number. Next, we study the power graph of a group and observe that non-isomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs should be isomorphic. It also is shown that the only finite
group whose automorphism group is the same as that of its power graph is the Klein group of order 4. We study the cozero-divisor graph of R denoted by ′(R) and we show that if ′(R) is a... 

Graph coloring is an important concept in graph theory. There have been much developments in this concept recently, vertex coloring, edge coloring and total coloring are studied. But one of the newest kind of graph coloring is [r; s; t] -coloring. This coloring is introduced in 2003 by A. Kemnitz and M. Marangio. In this coloring the difference of any two adjecent vertices, any two adjecent edges, and any adjecent vertex and edge must be at least r, s and t respectively. In this thesis we study [r; s; t] -coloring of graphs and discuss about its results  

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... 

Let R be a ring with unity, M be a unitary left R-module and I(M)* be the set of all non-trivial submodules of M. The intersection graph of submodules of M, denoted by G(M), is a graph with the vertex set I(M)* and two distinct vertices N and K are adjacent if and only if N\K ̸= 0. We investigate the interplay between the module-theoretic properties of M and the graph-theoretic properties of G(M). We characterize all modules for which the intersection graph of submodules is connected. Also the diameter and the girth of G(M) are determined. We study the clique number and the chromatic number of G(M). Among other results, it is shown that if G(M) is a bipartite graph, then G(M) is a star... 

A fullerene graph is a cubic and 3-connected plan graph that has exactly 12 faces of size5 and other faces of size 6, which can be regarded as the molecular graph of fullerene.In the irst part of this thesis we study some important deinitions and theorems whichused in the other parts.A matching of a graph G is a set M of edges of G such that no two edges of M sharean end-vertex; further a matching M of G is perfect if any vertex of G is incident with anedge of M. A matching M of G is maximum if |M| ? |N| for any other matching N in G. Amatching M is maximal if it is not a proper subset of some other matching in G. Obviously,any maximum matching in G is also a maximal matching. An...