Connectivity, Hamiltonian And Spectrum Of Mixed Cayley

The information is sweeping our world, the rapid development of informationtechnology has brought great convenience to human life in recent decades. ButComputer information network as an important carrier of global information, it is veryimportant. Involved in this field of graph theory to the study of information networks,the most important is the network reliability problem, The core issue of network reliabilityis the reliability analysis of communication networks and high-reliability network design,Graphas an effective model of the network topology, various properties of Graph have beenused to study the network reliability problem. But Cayley graph is a typical networkarchitecture model, it is a basis of research of the mixed Cayley graph, so the researchof the mixed Cayley graph is more significance. The three properties of the paper is tomeasure the classic parameters of the network reliability. Now Little research on ourcontury and abroad, so this paper has a high research value.In the first chapter, a brief introduction to the background of this thesis. In thesecond chapter, the mixed Cayley graph is the promotion and extension of the Cayleygraph and Bi-Cayley graphs, based the connectivity of method on Cayley graphs andthe Bi-Cayley graph, we obtain necessary and sufficient conditions for mixed Cayleygraph connectivity: Let G be a Abelian group,G₀=S₀,G₁=S₁,G=S₁22S₂,the Mixed Cayley graphX=MC （G, S₀, S₁, S₂）is connected if and only ifG=G₀G₁G2. In the third chapter, based on the Hamiltonian of Cayley graph insymmetry group and the Hamiltonian of Bi-Cayley graph in finite Abelian group. westudied the Hamiltonian of mixed Cayley graph in the cyclic group: let G be a cyclicgroup, G≥2, S_i G, i=（0,1,2），1G∈S₂,S_i^-1=S_i，whenS₀=S₁, then the connectedCayley graph MC （G, S₀, S₁,S₂）is Hamiltonian. In the fourth chapter, based on theadjacency matrix of the Cayley graph, we obtain the spectrum of the Cayley graph, acrossing tothe Cayley graph connect the Bi-Cayley graph, we obtain the spectrum of the Bi-Cayleygraph, using the method, we obtain the spectrum of the mixed Cayley graph: Let λ₁, λ₂,L,λ_nbe the eigenvalues of the Cayley graphC （G,S₀）,κ₁, κ₂, L,κ_nbe theeigenvalues of the Bi-CayleyBC （G,S₁）, If the adjacency matrix of C （G,S₀）andC （G,S₁）is normal and commute, then the eigenvalues of adjacency matrix of the MixedCayley graph areκ₁±λ₁,κ₂±λ₂, L,κ_n±λ_n.