Unimodality Problems On The Edge Cover Polynomials Of Some Graphs

Unimodality has been an important topic in combinatorics for a long time.It not only of great research significance,but is also closely related to other topics.For example,it often appears in computer science,economics,biology,and other fields.In recent years,the unimodality of graph polynomials has been widely studied,such as the independent polynomials and the chromatic polynomials associated with graphs.In this thesis,we consider another interesting class of polynomials,the edge covering polynomials of graphs.To study the unimodality problem of edge covering polynomials of some graphs,we first give the edge covering polynomials of graphs through the inclusion exclusion principle or the recursive relationship of polynomials,and then study the unimodal and logarithmic concavity of graphs.The details are as follows.The first part focuses on the research background and development of unimodal problems and edge covering polynomials,and briefly describes the research methods and main research contents of this thesis.In the second part,according to the recurrence relationship of polynomials,we first give the edge covering polynomials of complete graphs,and then prove the unimodality of edge covering polynomials of complete graphs in two parts and determine the location of mode.In addition,we establish a formula between the average edge covering polynomials of graphs of order n and the edge covering polynomials of complete graphs.It is proved that the average edge covering polynomials of graphs of order n are unimodal and can determine the location of mode.In the third part,the proof of logarithmic concavity of edge covering polynomials of some graphs is given.Firstly,according to the recurrence relation of polynomials,the edge covering polynomials of some graphs are given,including centipede graph W_n and its variation graph A_n¹,caterpillar graph H_n and its variation graph D_n,2¹,vertebrate graph V_n³ and its variation graph D_n,3¹,vertebrate graph V_n^t and its variation graph D_n,t¹.Then we prove that their edge covering polynomials are log-concave by means of Newton inequality.At the end of this thesis,by the recurrence relationship of polynomials,the edge covering polynomials of some graphs are given,including firecracker graph F_n³,graph H_2m+1and graph H_2m.