Genome sequencing is the process of determining in which order the nitrogenous bases also known as nucleotides within a DNA molecule are arranged. Every organism’s genome consists of a unique sequence of nucleotides....Genome sequencing is the process of determining in which order the nitrogenous bases also known as nucleotides within a DNA molecule are arranged. Every organism’s genome consists of a unique sequence of nucleotides. These nucleotides bases provide the phenotypes and genotypes of a cell. In mathematics, Graph theory is the study of mathematical objects known as graphs which are made of vertices (or nodes) connected by either directed edges or indirect edges. Determining the sequence in which these nucleotides are bonded can help scientists and researchers to compare DNA between organisms, which can help show how the organisms are related. In this research, we study how graph theory plays a vital part in genome sequencing and different types of graphs used during DNA sequencing. We are going to propose several ways graph theory is used to sequence the genome. We are as well, going to explore how the graphs like Hamiltonian graph, Euler graph, and de Bruijn graphs are used to sequence the genome and advantages and disadvantages associated with each graph.展开更多
The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. A graph is called perfect matching compact（shortly, PM-compact）, if its perfect matching polytope...The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. A graph is called perfect matching compact（shortly, PM-compact）, if its perfect matching polytope has diameter one. This paper gives a complete characterization of simple PM-compact Hamiltonian bipartite graphs. We first define two families of graphs, called the H2C-bipartite graphs and the H23-bipartite graphs, respectively. Then we show that, for a simple Hamiltonian bipartite graph G with ｜V（G）｜ ≥ 6, G is PM-compact if and only if G is K3,3, or G is a spanning Hamiltonian subgraph of either an H2C-bipartite graph or an H23-bipartite graph.展开更多
[App1.Anal.Discrete Math.,2017,11(1):81-107] defined the A_α-matrix of a graph G as A_α(G)=αD(G)+(1-α)A(G),where α∈[0,1],D(G) and A(G) are the diagonal matrix of degrees and the adjacency matrix of G,respectivel...[App1.Anal.Discrete Math.,2017,11(1):81-107] defined the A_α-matrix of a graph G as A_α(G)=αD(G)+(1-α)A(G),where α∈[0,1],D(G) and A(G) are the diagonal matrix of degrees and the adjacency matrix of G,respectively.The largest eigenvalue of A_α(G)is called the A_α-spectral radius of G,denoted by ρ_α(G).In this paper,we give an upper bound on ρ_α(G) of a Hamiltonian graph G with m edges for α∈[1/2,1),and completely characterize the corresponding extremal graph in the case when m is odd.In order to complete the proof of the main result,we give a sharp upper bound on the ρ_α(G) of a connected graph G in terms of its degree sequence.展开更多
Let G be a finite group, and S be a subset of G. The bi-Cayley graph BCay（G, S） of G with respect to S is defined as the bipartite graph with vertex set G x {0,1} and edge set {（g,0）, （gs, 1）1 g ε G, s εS}. In...Let G be a finite group, and S be a subset of G. The bi-Cayley graph BCay（G, S） of G with respect to S is defined as the bipartite graph with vertex set G x {0,1} and edge set {（g,0）, （gs, 1）1 g ε G, s εS}. In this paper, we first provide two interesting results for edge-hamiltonian property of Cayley graphs and bi-Cayley graphs. Next, we investigate the edge^hamiltonian property of F = BCay（G, S）, and prove that F is hamiltonian if and only if F is edge-hamiltonian when F is a connected bi-Cayley graph.展开更多
The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamilt...The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.展开更多
文摘Genome sequencing is the process of determining in which order the nitrogenous bases also known as nucleotides within a DNA molecule are arranged. Every organism’s genome consists of a unique sequence of nucleotides. These nucleotides bases provide the phenotypes and genotypes of a cell. In mathematics, Graph theory is the study of mathematical objects known as graphs which are made of vertices (or nodes) connected by either directed edges or indirect edges. Determining the sequence in which these nucleotides are bonded can help scientists and researchers to compare DNA between organisms, which can help show how the organisms are related. In this research, we study how graph theory plays a vital part in genome sequencing and different types of graphs used during DNA sequencing. We are going to propose several ways graph theory is used to sequence the genome. We are as well, going to explore how the graphs like Hamiltonian graph, Euler graph, and de Bruijn graphs are used to sequence the genome and advantages and disadvantages associated with each graph.
基金Supported by the National Natural Science Foundation of China under Grant No.11101383,11271338 and 11201432
文摘The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. A graph is called perfect matching compact（shortly, PM-compact）, if its perfect matching polytope has diameter one. This paper gives a complete characterization of simple PM-compact Hamiltonian bipartite graphs. We first define two families of graphs, called the H2C-bipartite graphs and the H23-bipartite graphs, respectively. Then we show that, for a simple Hamiltonian bipartite graph G with ｜V（G）｜ ≥ 6, G is PM-compact if and only if G is K3,3, or G is a spanning Hamiltonian subgraph of either an H2C-bipartite graph or an H23-bipartite graph.
文摘[App1.Anal.Discrete Math.,2017,11(1):81-107] defined the A_α-matrix of a graph G as A_α(G)=αD(G)+(1-α)A(G),where α∈[0,1],D(G) and A(G) are the diagonal matrix of degrees and the adjacency matrix of G,respectively.The largest eigenvalue of A_α(G)is called the A_α-spectral radius of G,denoted by ρ_α(G).In this paper,we give an upper bound on ρ_α(G) of a Hamiltonian graph G with m edges for α∈[1/2,1),and completely characterize the corresponding extremal graph in the case when m is odd.In order to complete the proof of the main result,we give a sharp upper bound on the ρ_α(G) of a connected graph G in terms of its degree sequence.
基金Supported by Natural Science Foundation of Inner Mongolia(2010MS0113)Inner Mongolia Normal University graduate students'research and Innovation fund(CXJJS11042)
基金partially supported by the NSFC(No.11171368)the Scientific Research Foundation for Ph.D of Henan Normal University(No.qd14143 and No.qd14142)
文摘Let G be a finite group, and S be a subset of G. The bi-Cayley graph BCay（G, S） of G with respect to S is defined as the bipartite graph with vertex set G x {0,1} and edge set {（g,0）, （gs, 1）1 g ε G, s εS}. In this paper, we first provide two interesting results for edge-hamiltonian property of Cayley graphs and bi-Cayley graphs. Next, we investigate the edge^hamiltonian property of F = BCay（G, S）, and prove that F is hamiltonian if and only if F is edge-hamiltonian when F is a connected bi-Cayley graph.
文摘The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.
