Sharif Digital Repository

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... 

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... 

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... 

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 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... 

Let G be a graph. A twin edge k-coloring of G is a proper edge coloring of G with the elements of Z_k so that for every vertex u and v of G we have s(u)≠s(v), where s(u) is the sum of all colors of the edges incident with u. The minimum k for which G has a twin edge k-coloring is called twin chromatic index of G and denoted by χ_t^' (G). In this thesis we find the chromatic index of paths, cycles, complete graphs, complete bipartite graphs and some complete tripartite graphs. In 2014 it was conjectured that if G is a connected graph with at least 3 vertices and maximum degree Δ(G), then χ_t^' (G)≤Δ(G)+3  

Let G be a graph. We call G an odd graph if all its vertices have odd degrees. Caro conjectured that for every graph G and every integer k, k > 2, there exists a Zk-coloring for the vertices of G so that for all v 2 V (G), the sum of the colors of all vertices in N[v] is not congruent to 0 modulo k. This thesis is mainly devoted to determining a lower bound for the number of vertices of the largest odd induced subgraph of a given graph with no isolated vertices. Another focus of this thesis is to find an upper bound for the number of odd induced subgraphs, as well as odd induced forests, needed to partition V (G), where G is a given graph with even order. At the end, two variations of Caro’s... 

In this thesis we investigate different factors of graphs with emphasis of even and odd factors. First, we prove the best existing lower bound for the largest even factor of graphs which is of the size at least 4/7|E(G)|+1 and we show that this bound is tight and classify the graphs which exactly meet this bound[]. Then we present a new lower bound for the largest odd-factor of graphs of even order which is of the size at least 2/5|E(G)|. We prove this bound using a new edge coloring called 1/2-coloring in which edges are colored with blue and red such that in each cycle the number of red edges is bigger than the number of blue ones. We prove in every 1/2-coloring of graphs with δ(G)≥3 or... 

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... 

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 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... 

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... 

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... 

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... 

The spectrum of a graph is related to many important combinatorial parameters. Let (G), ′(G) be the maximum number of edge-disjoint spanning trees and edge-connectivity of a graph G,respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of (G), we use eigenvalue interlacing for quotient matrix associated to graph to get the relationship between eigenvalues of a graph and bounds of (G) and ′(G). We also study the relationship between eigenvalues and bounds of (G) and ′(G) in a multigraph G. In the first chapter we prove eigenvalue interlacing and give several applications of it for obtaining bounds for characteristic numbers of... 

Let G be a graph of order n. The signless Laplacian matrix or Q-matrix of G is Q(G)=D(G)+A(G), where A(G) is the adjacency matrix of G and D(G) is diagonal degree matrix of G. The signless Laplacian characteristic polynomial or Q-polinomial of G is QG(x)=det(xI-Q(G)) and its roots are called signless Laplacian eigenvalues or Q-eigenvalues of G. If R is vertex-degree incidence matrix of G, then Q(G)=RRt. So Q(G) is a positive semi-definite matrix, i.e. its eigenvalues are none-negative. Let q1(G)≥q2(G)≥…≥qn(G) denote the signless Laplacian eigenvalues of G. A theory in which graphs are studied by means of eigenvalues of Q(G) is called signless Laplaciian theory or Q-theory.In this research,...