Sharif Digital Repository

Let X = fx1; : : : ; xng be the set of vertices of a hypergraph H and E = fe1; : : : ; emg be the set of its edges. The incidence matrix of hypergraph H is an n m and (0; 1)-matrix, N, such that nij = 1 if xi 2 ej and nij = 0, otherwise. A zero-sum flow for H is a vector with non-zero entries that is in the null space of N. A zero-sum k-flow is a flow with non-zero integer entries that absolue value of each entry is at
most k 1. In this thesis we study the existence of zero-sum flows for designs and hypergraphs. Also we consider zero-sum 3-flows in 2-Steiner designs  

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... 

Graph coloring is a fundamental problem in Computer Science. Despite its status as a computationally hard problem, it is still an active area of research. Depending on the specific area of requirement or an pplication, there are several variants of coloring.One such variant of graph coloring is Acyclic Graph Coloring. As with the case of proper coloring of graphs, Acyclic coloring can either be on vertex set or the edge set of a graph. When such a coloring is performed on the edges of a graph, it requires that after a graph has been colored there should not exist any cycle that runs on edges that are colored by only two colors. Such a cycle is called a bichromatic cycle. When a coloring does... 

Let G be a graph. A P3-factor of graph G is a subgraph of G such that each component is isomorphic to P3. In 1985 Akiyama and Kano conjectured that every 3-connected 3-regular graph of order divisible by 3 has a P3-factor. In this thesis we conjecture that every 3-connected 4-regular graph of order divisible by 3 has a P3-factor and also show that this conjecture concludes the previous one. Moreover, we investigate connected factors and some 3-vertex induced factors in graphs and show that every r-regular graph of order n (3 j n) has a P3-induced factor, if r _> 8n/ 9-1  

The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. Gutman et al. proved that for every cubic graph of order n, E(G) ⩾ n. Here, we improve this result by showing that if G is a connected subcubic graph of order n, n ⩾ 8, then E(G) ⩾ 1.01n. Also, we prove that if G is a traceable subcubic graph of order n,then E(G) ⩾ 1.1n. Let G be a connected cubic graph of order n, it is shown that E(G) > n + 2, for n ⩾ 8 and we introduce an infinite family of connected cubic graphs whose for each element, say G, E(G) ⩾ 1.24n, and some important conjectures will be raised about this. At the end, for a graph G and its vertex induced subgraphs H and K,... 

A flow for a graph or a combinatorial design is an element of the nullspace of its incidence matrix. For an integer r ≥ 2, an r-flow is a flow whose values belong to {±1;±2; : : : ;±(r - 1)} and for a real number r ≥ 2, a circular r-flow is a flow whose values are real numbers in the set [-(r- 1),-1]U[1, r - 1].In the context of undirected graphs, flows are called zero-sum flows. In the first part of this dissertation, we study circular zero-sum flows of graphs and prove a necessary and sufficient condition for the existence of a circular zero-sum r-flow in a graph in terms of its factors. Also, we investigate the minimum value of r for which a k-regular graph has a circular r-flow and... 

A signed graph is a graph with a sign attached to each edge. This article extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs.In particular, the largest Laplacian eigenvalue of a signed graph is investigated,which generalizes the corresponding results on the largest Laplacian eigenvalue of a graph.It is proved that (C2n+1; +) is uniquely determined by its Laplacian spectrum (or is DLS), where (C2n+1; +) is a signed cycle in which all edges have positive sign. On the other hand, we determine all Laplacian cospectral mates of (C2n; +) and hence (C2n; +) is not DLS. Also, we show that for every positive integer n, (Cn;) is DLS. Then, we study the spectrum of... 

Let G = (V(G),E(G)) be a Himple graph. An independent Het in a graph iH a et. of vert.iceH no t\vo of thE-'m are adjacE-'nt. The C'ardinality of a rnaxirnnrn independE-'ut fet in agraph G is r-ailed the independence number of G and is denoted b:y a:( G). The ndependencE- polynomial of G, I(G, :r), iH dE-'fined a l:ill'l I(G.1:) = ik(G)x'l \vhere iA.(G) is the number of iwh-•pendent ::->etH of G of Hize k and io(G) = 1. Then-' is nother graph polynomial \Vhir-h is called the domination poJ:ynomial and is defined as·ollmvs. A dominating sE-'t of G is a set 8 of vertices of G so that every vertex of G is fither in 5 or adjacent to a vertex in S. The domination... 

A signed graph is a pair like (G; ), where G is the underlying graph and : E(G) ! f1; +1g is a sign function on the edges of G. Here, we study the spectral determination problem for signed n-cycles (Cn; ) with respect to the adjacency spectrum and the Laplacian spectrum. In particular we prove that signed odd cycles and unbalanced even cycles are uniquely determined by their Laplacian spectrums, but balanced even cycles are not, and we find all L-cospectral mates for them. Moreover, signed odd cycles are uniquely determined by their spectrums but the signed even cycles, (except (C4;) and (C4; +)), are not and we find almost all cospectral mates for them. A mixed graph is obtained from a...