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#### Bromine-lithium Exchange Reaction in Organic Synthesis Using Glass Microreactor

, M.Sc. Thesis Sharif University of Technology ; Mohammadi, Aliasghar (Supervisor)
Abstract

Today, microreactors come into focus due to their exclusive characteristics including high surface-to-volume, lower consumption of reagents and fast mixing. In this study, optimization of the Br-Li exchange reaction as a consecutive reaction using glass microreactor is investigated. To construct glass microreactor, laser ablation and thermal bonding method are utilized. In order to find the best yield and selectivity of the reaction, various parameters, such as solvent, equivalent ration of the two reagents, concentration of the reagents, residence time and flow rates of the reagents are examined. In addition, atomic force microscopy (AFM) analysis was used to determine the surface...

#### Tutte Conjecture on Nowhere-zero Flow

, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract

In this thesis, the notion of nowhere-zero flow on graphs in defined with values is an Abelian group. It is shown that the existence of nowhere-zero flow with values in H depends on only the cardinal of H. Also the existence of nowhere-zero flowwith values on Zk is equivalent to the existence of the nowhere-zero flow with values in the set 1,2,...,k-1 on group Z.It is a easily seen that the nowhere zero flow on the graph with bridge doesn’t exist. Also has been proved any graph with no bridge has nowhere-zero 6 flow.Concidentally, the existence of nowhere-zero 5 flow for graphs with no bridge isstill an open problem.In 3-flow case, recently it is shown that any 8-edge connected graph has...

#### The Laplacian Spectrum of Graphs

, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract

In this thesis we investigate the spectrum of the Laplacian matrix of a graph. Although its use dates back to Kirchhoff, most of the major results are much more recent. The first chapter of this thesis is devoted to the integral Laplacian eigenvalues of graphs. In Section 2, particular attention is given to multiplicities of integer eigenvalues and to the effect on the spectrum of various modifications. In Section 3, the Laplacian integral graphs are investigated. The Section 4 relates the degree sequence and the Laplacian spectrum through majorization.The second chapter presents the result on permanent of the Laplacian matrix of graphs and permanental roots. In Section 2, we investigate...

#### On the Domination Polynomial of Graphs

, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)#### Some Applications of Combinatorial Nullstellensatz in Graph Theory and Combinatorics

, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)####
On Some Graph Theoretic Properties of Fullerenes

,
M.Sc. Thesis
Sharif University of Technology
;
Akbari, Saeed
(Supervisor)
Abstract

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...

#### Probabilistic Methods in Graph Coloring

, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract

Graph Theory and Combinatorial Analysis like the other branches of science use the probability for solving their problems. At the first , we will introduce the most common tools from probability used in Discrete Mathematics, such as “The Lovasz Local Lemma”. Most of them are based on Linearity of Expectation, Concentration Theorems and some other innovative methods such as Deletation Method. We will present different examples for these techniques. Also we will introduce “Random Graphs” and their importance. Specially we will explain how to use “Threshold Functions” to obtain different properties about the majority of graphs. Rainbow Connection, is a natural and interesting quantifiable way...

#### Intersection Graph

, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract

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...

####
Zero-sum Flows in Graphs

,
M.Sc. Thesis
Sharif University of Technology
;
Akbari, Saeed
(Supervisor)
Abstract

For an undirected graph G, a zero-sum flow is an assignment of non-zero real numbers on the edges of G such that the total sum of all edges incident with any vertex of G is zero. A zero-sum k-flow for a graph G is a zero-sum flow with labels from the set {±1, . . . , ±(k ? 1)}. In this thesis for a graph G, a necessary and sufficient condition forthe existence of zero-sum flow is given. It has been conjectured that if a graph G has azero-sumflow, then it has a zero-sum 6-flow. Also, it has been proved that this conjectureand Bouchet’s Conjecture for bidirected graphs are equivalent. It is shown that the con-jecture is true for 2-edge connected bipartite graphs, and every r-regular graph...

#### The Regular Graph Retated to Rings

, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract

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...

#### Weight Choosability of Graphs

, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract

Suppose the edges of a graph G are assigned 3-element lists of real weights. Is it possible to choose a weight for each edge from its list so that the sums of weights around adjacent vertices were different?

A graph G = (V;E) is called (k; k′)-total weight choosable if the following holds: For any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k′ real numbers, there is a mapping f : V [ E ! R such that f(y) 2 L(y) for any y 2 V [ E and for any two adjacent vertices x; x′

Σ e2E(x) f(e) + f(x) ̸=Σ e2E(x′) f(e) + f(x′)

Is it possible every graph is (2, 2)-total weight choosable and every...

A graph G = (V;E) is called (k; k′)-total weight choosable if the following holds: For any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k′ real numbers, there is a mapping f : V [ E ! R such that f(y) 2 L(y) for any y 2 V [ E and for any two adjacent vertices x; x′

Σ e2E(x) f(e) + f(x) ̸=Σ e2E(x′) f(e) + f(x′)

Is it possible every graph is (2, 2)-total weight choosable and every...

#### Graphs Associated with Algebraic Structure

, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract

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...

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...

#### Cayley Graphs and Annihilating-Ideal Graph of a Ring

, Ph.D. Dissertation Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract

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...

#### Path and Cycle Factors in 3-Regular Graphs

, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract

Given a graph G and a set F of connected graphs, an F-packing of G is a subgraph of G whose components are isomorphic to one member of F. In addition, if H is a subgraph of G, then an H-packing is defined similarly. The maximum F-packing is an F-packing such that it has the maximum number of vertices. If the F-packing F is a spanning subgraph of G, then F is called an F-factor. If each member of F is a path of order at least two (cycle), then an F-factor is called a path (cycle) factor. In this thesis, the focus was on the path factor and cycle factor in 3-regular graphs and these factors were investigated in 2-connected graphs, 3-connected graphs and bipartite graphs. Moreovere special...

#### On the Roman Domination Number of Graphs

, Ph.D. Dissertation Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract

Let G be a graph. A labeling f : V (G) ! f0; 1; 2g is called a Roman dominating function, if every vertex u with f(u) = 0 has at least a neighbor v with f(v) = 2. Define the weight of a Roman dominating function f to be w(f) =Σv2V (G) f(v). The Roman domination number of G is R(G) = minfw(f) : f is a Roman dominating functiong. Some other parameters are defined based on Roman domination number. A Roman bondage number bR(G) of G is the minimum cardinality of all sets E E(G) for which R(G E) > R(G). The edge Roman domination number of G, LR(G), is defined as R(L(G)), where L(G) is the line graph of G. In this thesis, after determining the exact value of the Roman bondage number for some...

#### Path Factors in Graphs

, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract

Let G be a graph. A path factor of a graph G is a family of distinct paths with at least two vertices which forms a partition for the vertices of G. For a family of non-isomorphic graphs, F; an F-packing of G is a subgraph of G such that each of its component is isomorphic to a member of F. An F-packing P of G is called an F-factor if the set of vertices in graph G and P are the same. The F-packing problem is the problem of finding an F-packing having the maximum number of vertices in G. In graph theory packing of the vertices of paths, cycles and stars are interesting subjects . This thesis is devoted to determine the conditions under which graph G has a {Pk}-factor, where by Pk we mean a...

#### On the Antimagic Graphs

, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract

A labeling of a graph is a bijection of edges in graph G to the set {1,2,…, m}. A labeling is antimagic if for any distinct vertices u and v, the sum of the labels on edges incident to u is different from the sum of the labels on edges incident to v. We say a graph is antimagic if it has an antimagic labeling.. In 1990, Hartsfield and Ringel conjectured that every connected graph other than K2 are Antimagic.In this thesis, we show that each graph with at least two degrees can be called Antimagic. We prove this conjecture for regular graphs of odd degree. and then it will be shown that Cartesian graphs have the property of Antimagic Labeling. Finally, we purpose a novel method for k-th powers...

#### Total Domination Number in Graphs

, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract

In this dissertation we aim to survey the concepts and results concerning Total Domination Set. Consider graph G = (V;E). Let S be a subset of V . S is Total Domination Set in G if every vertex in V is adjacent to at least one vertex in S.Furthermore, we call S a k-Total Domination Set if every vertex in V is adjacent to at least k vertices in S. The size of the smallest Total Domination and k-Total Domination Set in a graph G is respectively called the Total Domination and k-Total Domination Number of G.Now, if there is a sequence of subsets of V like S = V0; V1; V2; : : : ; Vl = V so that for each i, every vertex in Vi is connected to at least k vertices in Vi1, S is a Dynamical k-Total...