Research On The Permanental Polynomials And The Characteristic Polynomials Of Octagonal Chains

Graph theory mainly studies the internal structure contained,in graphs,in which the research on the polynomials of graphs is an important field.It mainly character-izes the structural properties of graphs through the parameters of matrices associated to their graph representations,and studies the intrinsic relations between the param-eters and structure of graphs.An octagonal system is a 2-connected graph consisting with some regular oc-tagons of unit edge length.An octagonal chain is an octagonal system if it has no vertex belonging to three octagons and no octagon with more than two adjacent octagons.In this thesis,according to the permanental polynomials and the charac-teristic polynomials theory of graphs,the extremal octagonal chains with n octagons having the minimum(resp.maximum)coefficients sum of the permanental polyno-mial and spectral radius are identified,respectively.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 first give some necessary notations and terminologies,then introduce the construction of octagonal chains and a roll-attaching operation on the octagonal chains.· In Chapter 3,by studying the permanental polynomials of graphs,we determine the effect on the coefficients sum of the permanental polynomials of octagonal chains under the roll-attaching operation.And then the extremal octagonal chains with n octagons having the maximum and minimum coefficients sum of the permanental polynomial are identified,respectively.· In Chapter 4,based on the characteristic polynomials of graphs,we study the influence on the spectral radius of octagonal chains under the roll-attaching operation.Furthermore,we characterize the extremal octagonal chains with n octagons having the maximum and minimum spectral radii.· In Chapter 5,we summarize the main results in this paper and give some prospects for further research in the future.