| Let Gw be a weighted graph. The number of the positive, negative and zero eigenvalues in the spectrum of Gw are called positive inertia index, negative inertia index and nullity of Gw, denoted by i+(Gw), i-(Gw) and i0(Gw), respectively. The rank of A(Gw) is called the rank of Gw, and denoted by R(Gw). Obviously, R(Gw)= i+(Gw)+i-(Gw). This paper which stands on the basis of previous results, do further researches on inertia index and rank of weighted bicyclic graphs and the weighted graphs with given acyclomatic number which has a special kind of graph as its induced subgraph. The concrete content is in the following:◠In Chapter 1, we introduce the background and significance of the research, including the development of a representative at home and abroad regarding this aspect. Based on this research background and profound discussion, by using deep-going analysis, it fully shows the main work’s necessity and innovation.◠In Chapter 2, we give some necessary definition and lemmas.◠In Chapter 3, we characterize all weighted bicyclic graphs having exactly one or two positive (resp. negative) eigenvalues and we characterize all weighted bicyclic graphs of rank 2,3,4.◠In Chapter 4, we characterize all weighted graphs with given acyclomatic num-ber which has a special kind of graph as its induced subgraph of rank 2,3,4 and we characterize those graphs having exactly one or two positive (resp. negative) eigenvalues.◠In Chapter 5, we summary the paper and give prospects in the future. |
